Do many Americans identify with other countries in terms of ancestry, rather than Britain?

Yes. America is one of the most (if not the most) ethnically diverse country in the world. We have citizens that are from just about every other country. Almost all citizens are descended from some form of immigration to America. Not every white American descended from the British. As for being allowed to, what do you mean? You think that being an American means that you give up your heritage? I admit, that most African Americans wouldn't know exactly what country in Africa that their ancestors were from, being that they were taken as slaves. And the Native Americans are the only group of people who aren't descended from immigrants.