Again, I would argue that we do have to choose. I also think that the tension between these positions is what is causing quite a bit of conflict in American politics right now.
There is an essentialism inherent with these contexts. Just like French"ness" or British"ness", we have white"ness" or black"ness", and it can be applied to any race, sexual preference, or gender identity, etc., etc. This view has practical limits, and where those are drawn is important, because we run into trouble with the "green eyes"-ness brought up by the previous poster. And, there is ultimately a conflict that comes from this.
Non-essentialists would argue that the attributes that make up these differences are effectively accidental, so eliminating the differences is arbitrary, and perfectly reasonable. This means that nothing is lost when one context is adopted by the other. Different people have different views on this though. This also means that for the non-essentialist, negative-based differences must also be accidental, which is an important difference, and while essentialists may want to eliminate the negative-based differences, there is a sense in which the essentialist view endorses the idea that they must exist.
Practically speaking though, when we have people participating outside of their context, does the essentialism evaporate? Should it? That is to say, if white people start moving into a Chinatown, does it ever stop being an "Asian neighborhood"? At what point does it stop? Should we try to preserve it? If we do, is it right to discriminate in doing that?
This is the conflict between acculturation, assimilation, and amalgamation? Essentialist views often see mass adoption as acculturation. Non-essentialists see it as amalgamation. These are subtle differences, but they yield vastly different policy prescriptions.