North America and South America are continents. Most people call the country America and the people that live there Americans. Would you call a Canadian or a Mexican an American???
I am a Canadian, and I do call myself an American too. That is, a resident of the continent of America. Mexicans and others who live in the Latin part of America all consider themselves American, because in Spanish and Portuguese
América is a continent and therefore
americano isn't a nationality but a continental identity. Those languages have other words for the US national identity,
estadounidense in Spanish and
estadunidense in Portuguese. That's what people in Latin America learn in school.
This is because for a few hundred years after the voyages of Columbus and Amerigo Vespucci made America known to Europeans, everyone considered all of America to be one continent. It would be stupid for America to mean something different from
Amerika,
América,
Amérique, etc. since those are just different linguistic forms of the same word, and those other forms all refer to the whole continent in the western hemisphere. When North Africa is obviously in Africa and south Asia is obviously in Asia, it would also be stupid to say that North America and South America are not both in America.
I reject the arrogation of the words America and American by the US as being totally wrong. In Canada's two official languages I've adopted the English and French equivalents of
estadounidense and
estadunidense for the people, places and institutions of the United States, which are
Usonian and
états-unien and I only use American and
américain in their correct, continental sense.