Ethnicity is a social construct with some fuzzy boundaries, but I don't think anyone credible tries to claim that there is an "American Ethnicity". Usually when that term comes up it's from some racist overly proud that someone in their ancestry came over on the Mayflower.

Personally I think it's one of the strengths of this country that a first generation immigrant can come here and become an American. I don't think this is very common around the world.