We all project traits on groups of people. It's how we operate socially, on a large-scale level, and also on a tiny, day-to-day, individual level. If we did not have a personal stable of stereotypes established over many years of interaction, we would barely be able to interpret anything at all.
Modern example:
The defining trait of the group "people in Saudi Arabia" is their location, not their appearance. But is it bigoted to say that people in Saudi Arabia are highly religious? They sure are tolerant of theocracy there, though I bet if they were more connected to the outside world and less dependent on welfare from kings, they would rise up. If I meet someone from Saudi Arabia I am fully prepared to assume they pray to Allah five times a day, and are super into their religion, relative to me. Does that assumption make me more likely to cause a faux pas, or less?
Historical example:
The Vikings that inhabited Iceland for hundreds of years were socially conservative, environmentally destructive, and relatively violent in their settling of disputes. Written accounts all over Europe and from within Viking society, as well as clear evidence that Iceland was divested of 97 percent of its trees after they arrived, back these claims. The majority of male settlers came from the Nordic countries, largely Norway, but most female settlers came from the British Isles, kidnapped by swordpoint. One could make the case that half their genetic legacy is borne of rape. Their history is loaded with warring chiefs and raids, and only settled down after they decided to swear allegiance to a foreign king and take up Christianity, and basically stopped being Vikings.
If I was cast back in time to 900 AD and wandered around Iceland, assuming I could speak their language, would that knowledge help me? Probably not. Having no family affiliation, I'd be seized as a slave, and probably die of exposure or starvation in short order.
One might assume that all native Icelanders today are descended from Vikings, and that therefore the stereotypes apply. But since the time of Vikings, the population was ravaged by plague and volcanism and even plunder from English merchants, all multiple times. Genetically the current population is barely distinct from the rest of Northern Europe. And yet, I found that two of the three stereotypes held true when I traveled there: Icelanders are definitely not violent, but they remain socially conservative, and they continue to be environmentally destructive (though they have the grace to be ashamed of it and are trying to change course.)
Back in the day, everyone considered "race" and "geographical origin" interchangeable. And for understandable reasons, what with how limited travel was. In modern times we've driven a wedge between these things and started to whittle down the importance of "race" as a carrier of behavior and value. But there is still so much progress to make, clearly, because people with the same geographical origin but a different appearance are still treated very differently, within their own communities as well as beyond. The problem is not embedded in genes and skins, but in minds and media. To work against stereotypes you sometimes need to know in detail what they are, but by deciding to work against them, you are also forced to act upon them. It's difficult and complicated work.
If I meet a Black man on the street in Oakland, I bring to bear a decades-long accumulation of assumptions about how that man perceives me, how other people who look like me have treated him, and how I can present myself so as to show I am not bound by those assumptions and will treat him with dignity and camaraderie.
By moderating my behavior with this knowledge, am I engaging in racist behavior? Would it be better for both of us if I was completely unaware of any stereotypes held by society at large, like my young nephews generally are? I would love for there to be a clear answer to this, but: A little bit yes, a little bit no. My take on it is: It's very likely I can be more helpful to that man by knowing what we're both working against. And that we both need to be a little more self-aware than usual in our first meeting because of it all.
An important question to ask is, what's the victory condition here? Obviously, the stereotypes we want to fight in modern society are negative ones based on appearance alone, and second to that the negative ones based on place of origin. Even though, like in the examples above with Vikings and Saudis, they may even contain truth. Do we need to erase the distinction in order to erase the stereotype? Should we all aspire to "not see color"? Or should we aspire to make the color we do see irrelevant when it comes to opportunity and trust, perhaps by putting our thumb on the scales of justice?
Almost all the signifiers of the various Germanic and Danish and Irish origins of my motley family were rapidly crushed or abandoned in just a few generations, because they were a liability. No distinction, and there's nothing to hang a stereotype on, and that means safety. It would have been nice to keep those distinctions. It would have been nice to arrive in a land where there was a certain amount of embracing going on to complement the rejection. Somewhere in the middle is a territory called assimilation. How do we expand that territory?
I don't think it's a process of embracing everything. I don't think the answer is "accepting multiple cultures in one society". That's another way of saying "everyone stays in their assigned seating and doesn't get to make judgements or change." I think it's a process of sorting out the differences that matter from the ones that don't, which requires a collective agreement on the importance of principles over customs, and some relaxation in possessiveness over one's own cultural identity. Definitely not too much, but some. And damn right it's complicated, because people use external signifiers - like clothing, prayers, gestures, phrases, ceremonies of all kinds - to advertise and reinforce what they believe in, and those things are easily misinterpreted. Like the abaya for example. Or a prominently worn cross. Or saying "God bless you" to an atheist.
This is interesting to me because I suspect that people around my own political spectrum - left-leaning liberals in California - don't actually want multiple cultures coexisting. What they want, is one unified culture that embraces liberal principals, but also displays all the outward trappings of cultures - including far less tolerant ones - from all over the world. People wearing abayas and carrying prayer mats, happily standing shoulder-to-shoulder with people wearing long pants and kippot, right next to people wearing gold glitter mankinis and shouting "yas queeen!", with no sense of discomfort, because deep down they have all embraced the same liberal values, and keep their distinct appearance only because it feels fashionable or comfortable for them.
That's an odd sort of utopia, but not an unpleasant one. The trouble is that people around me seem to believe that liberal values are so self-evident and obvious, that all they need to do when faced with - for example - a militant Islamic fundamentalist, is make him as comfortable as possible in their midst, and then he will change his mind about wanting to beat any woman who walks in public without wearing a burqa, because isn't diversity wonderful?
If I am a guest in a Christian household and they take me to church on Sunday, and the morning sermon contains a dozen references to how misguided and shameful homosexuals are (true story), what is the better move? Stand up and walk out, embarrassing the members of the household that are hosting me, but making a powerful example to the congregation that people who would object to this treatment are in their very midst? Or sit there and say nothing, because I am a guest and must be tolerant of their religious ways? Because I must be accepting of their culture?
Conversely, to a Christian, is the solution to this dilemma, "we will live among these homosexuals even though we know they are sinful and gross, because slaughtering them would be worse?" That does not strike me as progress, but rather a dead-end.
Liberal values are not just to be endorsed by example alone. They are to be insisted upon; transmitted through aid and commerce whenever possible. Just as a church proselytizes to the poor people who wander in to the soup kitchen, we should be pushing for equal treatment under the law, the acceptance of all consensual sexuality, free speech, and the liberation of the mind from the rule of gods and kings.
Modern example:
The defining trait of the group "people in Saudi Arabia" is their location, not their appearance. But is it bigoted to say that people in Saudi Arabia are highly religious? They sure are tolerant of theocracy there, though I bet if they were more connected to the outside world and less dependent on welfare from kings, they would rise up. If I meet someone from Saudi Arabia I am fully prepared to assume they pray to Allah five times a day, and are super into their religion, relative to me. Does that assumption make me more likely to cause a faux pas, or less?
Historical example:

If I was cast back in time to 900 AD and wandered around Iceland, assuming I could speak their language, would that knowledge help me? Probably not. Having no family affiliation, I'd be seized as a slave, and probably die of exposure or starvation in short order.
One might assume that all native Icelanders today are descended from Vikings, and that therefore the stereotypes apply. But since the time of Vikings, the population was ravaged by plague and volcanism and even plunder from English merchants, all multiple times. Genetically the current population is barely distinct from the rest of Northern Europe. And yet, I found that two of the three stereotypes held true when I traveled there: Icelanders are definitely not violent, but they remain socially conservative, and they continue to be environmentally destructive (though they have the grace to be ashamed of it and are trying to change course.)
Back in the day, everyone considered "race" and "geographical origin" interchangeable. And for understandable reasons, what with how limited travel was. In modern times we've driven a wedge between these things and started to whittle down the importance of "race" as a carrier of behavior and value. But there is still so much progress to make, clearly, because people with the same geographical origin but a different appearance are still treated very differently, within their own communities as well as beyond. The problem is not embedded in genes and skins, but in minds and media. To work against stereotypes you sometimes need to know in detail what they are, but by deciding to work against them, you are also forced to act upon them. It's difficult and complicated work.
If I meet a Black man on the street in Oakland, I bring to bear a decades-long accumulation of assumptions about how that man perceives me, how other people who look like me have treated him, and how I can present myself so as to show I am not bound by those assumptions and will treat him with dignity and camaraderie.
By moderating my behavior with this knowledge, am I engaging in racist behavior? Would it be better for both of us if I was completely unaware of any stereotypes held by society at large, like my young nephews generally are? I would love for there to be a clear answer to this, but: A little bit yes, a little bit no. My take on it is: It's very likely I can be more helpful to that man by knowing what we're both working against. And that we both need to be a little more self-aware than usual in our first meeting because of it all.
An important question to ask is, what's the victory condition here? Obviously, the stereotypes we want to fight in modern society are negative ones based on appearance alone, and second to that the negative ones based on place of origin. Even though, like in the examples above with Vikings and Saudis, they may even contain truth. Do we need to erase the distinction in order to erase the stereotype? Should we all aspire to "not see color"? Or should we aspire to make the color we do see irrelevant when it comes to opportunity and trust, perhaps by putting our thumb on the scales of justice?
Almost all the signifiers of the various Germanic and Danish and Irish origins of my motley family were rapidly crushed or abandoned in just a few generations, because they were a liability. No distinction, and there's nothing to hang a stereotype on, and that means safety. It would have been nice to keep those distinctions. It would have been nice to arrive in a land where there was a certain amount of embracing going on to complement the rejection. Somewhere in the middle is a territory called assimilation. How do we expand that territory?
I don't think it's a process of embracing everything. I don't think the answer is "accepting multiple cultures in one society". That's another way of saying "everyone stays in their assigned seating and doesn't get to make judgements or change." I think it's a process of sorting out the differences that matter from the ones that don't, which requires a collective agreement on the importance of principles over customs, and some relaxation in possessiveness over one's own cultural identity. Definitely not too much, but some. And damn right it's complicated, because people use external signifiers - like clothing, prayers, gestures, phrases, ceremonies of all kinds - to advertise and reinforce what they believe in, and those things are easily misinterpreted. Like the abaya for example. Or a prominently worn cross. Or saying "God bless you" to an atheist.
This is interesting to me because I suspect that people around my own political spectrum - left-leaning liberals in California - don't actually want multiple cultures coexisting. What they want, is one unified culture that embraces liberal principals, but also displays all the outward trappings of cultures - including far less tolerant ones - from all over the world. People wearing abayas and carrying prayer mats, happily standing shoulder-to-shoulder with people wearing long pants and kippot, right next to people wearing gold glitter mankinis and shouting "yas queeen!", with no sense of discomfort, because deep down they have all embraced the same liberal values, and keep their distinct appearance only because it feels fashionable or comfortable for them.
That's an odd sort of utopia, but not an unpleasant one. The trouble is that people around me seem to believe that liberal values are so self-evident and obvious, that all they need to do when faced with - for example - a militant Islamic fundamentalist, is make him as comfortable as possible in their midst, and then he will change his mind about wanting to beat any woman who walks in public without wearing a burqa, because isn't diversity wonderful?

Conversely, to a Christian, is the solution to this dilemma, "we will live among these homosexuals even though we know they are sinful and gross, because slaughtering them would be worse?" That does not strike me as progress, but rather a dead-end.
Liberal values are not just to be endorsed by example alone. They are to be insisted upon; transmitted through aid and commerce whenever possible. Just as a church proselytizes to the poor people who wander in to the soup kitchen, we should be pushing for equal treatment under the law, the acceptance of all consensual sexuality, free speech, and the liberation of the mind from the rule of gods and kings.