With a view toward dividing the country into black and white and pretending that no other ethnic identities exist, laws were passed during the formation of the U.S that isolated people of African descent by placing us outside the "mainstream." Everybody else, unless whites said no, was white and presumably a part of the mainstream. These laws and the unwritten customs that accompanied them operated to maintain the myth of white supremacy. They also made black the embodiment of all that is bad/evil/wrong, and of course white the opposite. Other "obvious ethnics" (see W. Kamau Bell's comic turn on this term) resent this black-white division with good reason.
One result of lumping all Europeans into the "white" category is that many of them no longer have any notion of their ethnic origins. And, in California I've observed that many could care less as more and more people ignore ethnic boundaries. I know one young man who is the product of an Italian-American mother and Mexican-American father. His wife is the product of a father who is Greek-American and a mother who is Korean-American. They have two children. How do we classify them? And, who cares?
At some point we may even stop using that ridiculous appellation, "bi-racial," usually reserved only for the offspring of one white parent and one obviously ethnic parent. Unless, of course, that offspring becomes the first black president of the United States.
At this point, as global interaction on every level increases, I think ethnic identification will become far less significant than it has been. That won't happen anytime soon, but things are certainly moving in that direction.