You've made a good point here. And yet I have to ask: why is it that people feel the need to take pride in their differences? The people who insist on making others acknowledge their heritage are likely doing it out of a since of pride. Does that not divide people? And why is it important for someone to be acknowledged as "African-American" or whatever?
The thing to remember is that the cultural differences between groups exists with or without overtly acknowledging them. Cultural differences are impossible to prevent because we only have a small amount of people that we can interact with directly. While humanity is less isolated than it has ever been before, the nature of humans to align with each other into smaller groups, or tribes, is a part of what has allowed us to survive and thrive as a species. We are social creatures and we
need to feel as though we belong to a pack or tribe.
But an ambiguously defined pack of "humanity as a while" doesn't satisfy this need. We need to feel that sense of belonging on a smaller, more concrete level. We create these alignments according to our cultural similarities, for the most part. Nowadays, in the absence of the extreme isolation of the past, those packs have the potential to reach much larger sizes than previously possible. People will align with a pack or packs based on their similarities to the other members of those packs. Shared history, shared language, cultural values, etc. Sometimes, those things coincide with shared physical characteristics, sometimes they do not.
In the case of self-identifying as an African-American, for example, there is the incidental aspect of shared physical characteristics, but the
real impetus for this identification is the shared cultural history, language, and values. The same is true for people who identify as Irish-American, or simply as an American, for that matter. Physical aspects are far less of a factor in these groupings than the cultural aspects are. think about it.
When you pick someone whom you wish to associate with, you can (and probably will) totally ignore physical characteristics. You'll probably focus more on their traditions, values, and the other social characteristics which the two of you may share. People generally don't enjoy associating with someone they have little to nothing in common with. They may not dislike the other person, but they aren't likely to bond with them without a decent amount of common ground either.
This is basically what causes that group formation. The differences have to exist
before the groupings can even occur. So giving name to the groups does not act as a divisive factor. The divisions pre-date the names for them.