Not to try too hard to explain the joke but I think the core concept being highlighted here is one of a perceived discrepancy between "diversity inclusive descriptors" and terms that imply "otherness." For example, a white person might feel uncomfortable using the term "black" but would be comfortable with terms like "person of color" and "African-American." Linguistically, this might be because "person of color" implies that the individual is first and foremost a person and that their color, in an ethnic sense, is an additive quality to their "personness." I'm a person. You're a person. We're all...persons. That sort of thing. Similarly, a person who is African-American is, much like the (I'm going to assume American) white speaker, also American. It's the idea of an immediately identifiable, if unspoken, shared quality.

There’s a similar bit from Schitt’s Creek I really like about the word “Jewish”. I’d post it here but I can’t find a good quality version