Who are the "real" Americans, anyway?

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So, the Americas were named that after Amerigo Vespuci. Neat, right? Columbus came here and thought it was India, or it was supposed to be. Just tidbits. But why do United States citizens refer to themselves as Americans, but everyone else in the American Continents are considered by their country: Canadians, Mexicans, Guatamalans, for example. Ok, so we've got the name United States of America, but in reality, in the rest of the world, a country is equivalent to a state (a lot of the time), so our name doesn't make all that much sense. But really, who are the "Americans" if both continents are America, and no one else claims the name, really?