First, "being an American" is a term usually used exclusively by people in the United States to refer to being a United States citizens. Canadians and Mexicans often see this usage as lacking cultural awareness, as there are three different nations in North American and many more in Central and South America. Being Bolivian, Guatemalan, Canadian, or Mexican is just as much a matter of being a citizen of one of the nations of the Americas as being born and living in the United States.
Whether you consider the Americas as a whole or only the United States, we are all immigrants. Even the First Nations (or Native Americans) originally traveled to the Americas from other parts of the world (human life, in fact originated in Africa). Thus our cultural identities are a blend of the cultures from which we originated (if the immigration was within our historical memories) and our current place of residence or citizenship.