This is not a matter of right or wrong, but perspective.
The point I am making is not supposed to be very specific, but more broad.
Of course if an American moves to a different country, he or she will carry and relate to American culutre... When did I ever say different?
Yes, there are differences, but the whole culture is not present. Even the small differences can be seen as part of the American culture.
Yes, some people can related to decending cultures well, but in general, most people do not know what what the other culture really is like. Sure you might know some traditions of your relatives, but I do not believe that this makes you "German" or what your ethnicity is.
I am not saying people are wrong for calling themselves a certain name, it is just one of my general perspectives.