I was born in St. Louis, MO and raised in New York City, and was brought up singing “America the Beautiful” and “God Bless America,”and was therefore conditioned to call myself an American and refer to my country as America.
I maintained this proclaimed identity until a friendly guy from Colombia gave me a logical lecture about the reality of America explaining that he too is an American because he lives on one of the American continents. He went on to explain that we who live in the US are referred to by Latin Americans as Norte Americanos (North Americans) or Estadounidenses, a Spanish way of saying a United States citizen.
I later attended a Latin-American cultural event hear in the U.S. where the speaker pulled up a black doll from her country referring to it as African-American. What really hit home for me was while traveling through South America I stopped in a bank to withdraw money using my credit card. The bank manager asked me of which country is my bank. Out of frustration, I said, “the United States, can't you see Bank of America on the card?” Her response was that could be anywhere in the Americas.
I think that it is somewhat arrogant of us here in the U.S. to label our one country out of two continents with the misnomer of America. However, people around the world also refer to our country as America. I now refer to my country as the United States of America or simply the USA, and when I'm speaking to Spanish speakers, I refer to myself as Estadounidense, not Americano.
America is no more a country than Africa, Asia, or Europe. They are all continents.