I think I'm about as American as they come. My last name may be from the Azores, but I don't speak Portuguese, and no tradition has been passed on from there that I'm aware of. I have ancestors from England, Scotland, and other places as well, but at least five generations of my family has lived in New England. I can even trace my lineage back to the Plymouth colony.
I'm an American, from an American family. If I trace back far enough, I could claim to be from a Portuguese family because of my name, but to do so would be to both ignore my diverse ancestry and to define my identity in terms of a culture I know little about. If I followed that as far as it could be taken, I may as well say I'm from the cradle of civilization for all the relevance it has to my life.
When I think of my roots, I think of fishermen, farmers, blacksmiths, mechanics, smugglers, pharmacists, teachers, lifesavers, racecar drivers, firemen, printers, and lighthouse keepers, all of whom were New England Yankees. Call me an American, a Yankee, or a mutt... my roots are not in far away lands.