Actually our name is not United States either, we are The United States of America. "United States," "the States," and yes even "America" are commonly used and understood nicknames.
People of the United States are most commonly referred to and understood as "Americans." If you say "I'm an American," virtually no one would think that you meant that you were vaguely from one of the two American continents, North or South. They would know that you meant you're a citizen of the United States; you know, America.
America != United States We know that an "United Stater" is also an American, but there are Americans who are not "United Staters".