I think it’s self-declared, and only the first ethnicity given is used.
If they went by last name, most African-Americans would be listed as English and there would be no AA pluralities.
I agree; and sometimes it has to do with the surrounding majority's point of view. When I lived in an Italian neighborhood, I was "Irish", because most of the other NW Europeans who lived there had been; but in Amish country I am "English." In reality I have ancestors from many European countries as well as native American; my European antecedents have been here for the most part of 400 years; and there is barely any predominance among those ethnicities. While there might be a Welsh or German surname, when you look back in the genealogy, the ancestors might have been Dutch, Irish, English, whatever. People move around. Either that, or Americans, as a nation of immigrants, are predominantly descended from energetic people who move around.
Up until 2009, I was always proud to say merely that I am an American.