Well, there's not really much else we could call ourselves. We were the first of the European colonies to form an independent country (of 13 separate states). Hence, the United States of America (itself a European name of the continent). The term American, in reference ot the citizens of the US is simply shorter than "a citizen of the United States of America."
I do agree with you, though, that we aren't the only ones with a right to be called "Americans" any more than (say) the Germans could be the only ones with a right to call themselves "Europeans". On the other hand, like I said, what else would we call ourselves?
Marc