I think a starting point for this is to think about the history of how it got this way.
Generally it's something like by some kind of luck, the reds became dominant. Perhaps they had a more ruthless leader, more resources or something... maybe they found a stash of ancient laser guns, maybe their magic is more suited to combat, doesn't matter what. The reds banded together because they thought of themselves as a "family" and they thrived through whatever advantage they had, that's the important part.
Then the children and descendents of those reds inherited that privileged position. They had better houses, better lands, and so a leg up in becoming educated, having a better life. They put rules into place to protect their children, like "nobody can hunt animals on red-owned land" (because it's their land!) and "nobody is allowed to steal property." (nobody likes their stuff getting nicked, even if they have loads of stuff and more than they need!) They probably set up some sort of guard or a warrior caste or cops to enforce these new laws.
The blues, and especially greys and purples were stuck out on the less fertile and habitable land, without access to resources, and would be punished if they resorted to stealing the plentiful food the reds have, or hunted on their vast lands. When petitioning the reds for help, the reds would perhaps say, "I don't see what the problem is, there's plenty of land. You just have to work hard, like we do."
Perhaps at some point, blues have attacked or raided the reds in desperation, (it probably resulted in failure or short-lived victories because the reds have all the resources for a better equipped, trained and fed army) and the reds might use this to justify the idea that "blues are violent. If they just protested peacefully and asked for help nicely maybe they'd get what they want".
And the red guard/knights/samurai/cops were seen as justified in beating the crap out of people who break laws, because "well, it's the law!" and probably started to attract bullies who like beating the crap out of people. Sometimes it would reach the ears of a "nice" red that a purple had been beaten to death... and that red would say "Oh... how horrible! Those particular police need to be punished!" because they frame it as a problem of "some people are racist and that's bad" and not "we have an unfair system that criminalises blueness and makes it easy for bigots to be in a position where they have an excuse to beat the crap out of blues on the flimsiest of reasons." or "Our monopolisation of resources forces blues into criminal activity out of desperation".
Because if you have a history like that, the bigotry becomes systemic, more like real bigotry. It's not just that some people have an illogical aversion to people who look different, it's that in an effort to protect family and cultural legacy for their children, the reds (even the nice ones) have pushed the other colours into a disadvantaged position... but then forgotten that their ancestors did that through violence or opportunism, because they personally haven't done anything like that. They're scared to lose what they have, because they like their lifestyle and want their kids to share that same lifestyle, and ceding some of their lands, opportunities or resources to the blues would mean they have less and their children have less. This applies the same way with say... being gay; the reason for the fear of it is based in a fear of the threat to family legacy in cultures that have grown around the idea that securing a good marriage and then offspring to inherit the "family name" is important, and somebody who just keeps running around having "unproductive" sex with same-sex partners is a threat to that, and may "tempt" others into that lifestyle. This is why even in societies that were tolerant of homosexuality, like say ancient Greece, feudal Japan, it was expected to be more like a "side activity", and those participating were still meant to also marry a woman and have kids. But fear of homosexuality and different gender expressions became so codified in culture and religion that to many people it becomes just a more general fear or revulsion of the act itself on principle. Sometimes it's easier to say "It's sinful!" than confront the real, practical, social reasons it was originally ruled as bad.
Generally the "problematic" way to depict bigotry is like "it's just something bad people do." "It's bullies calling people mean slurs" etc. And on the other side, it'll be like "There are violent blues, but they're the bad ones, the good blues attempt to get social change by peaceful protest or working their way up through the meritocratic system", and then showing everyone in schools and workplaces all treating them very nicely making it seem like any blues who protest violently or resort to crime are just random assholes. If that level of bigotry exists in a setting, you have to approach it from a base understanding that something is deeply wrong in society on a structural level.
It might be worth watch HBomberguy's "RWBY is Disappointing" video, because it explores exactly how RWBY makes a mess of trying to explore bigotry by taking a shallow approach where the racism seems to have sprung from nowhere, there's nothing wrong with society and the protesters are evil baddies the heroes are allowed to beat up because they all affect stereotypical villain behaviours....