Why do most U.S. Citizens believe that American is exclusive to them?

In common parlance, America is a country. There is no American continent. There is a North American continent and a South American Continent. A person from Canada is a North American. A person from Peru is a South American. If you can't understand this, then you might as well tell people in South Africa that they can't call themselves South Africans because there are other countries in Africa.