Fairly sure that most Americans of Japanese ancestry consider themselves both Japanese and American. Same goes for most other ethnicity. Does that make them any LESS American?

Many Puerto Ricans consider themselves both Puerto Rican AND American. This is especially interesting given Puerto Rico is an American territory yet it is considered a different cultural group.

Germans of Turkish descent are in fact still Turkish regardless of generation. Similarly Germans of Chinese descent are still Chinese while also being German.

Your final sentence is the nut. The rest of it is just discrimination (small d) of people into cultural groups.

(Edit: As an aside... Is Doner German culture? Or is it Turkish?)