It is true tha continents do not define citizenship, but it is normal to classify people from their continent of origin: European, African, Asian, American. Well, actually these are two continents, so better specifying North American or South American.You are a citizen of a country. There is no country called Central or South America. There is only one country in the world with America as part of it's name. Continents do not have citizenship. American is the proper term for somone from the United States of America. American isn't what people from the USA insist on, it's what the rest of the world insists on. Mexicans refer to us as Americanos.
United States and US is what is misplaced. Other countries have "states" as part of their name. Estados Unidos Mexicanos, for instance.
Here in Italy, we call Americano a guy from USA or perhaps Canada, and we assume he speaks English. Sudamericano instead from anywhere south of USA (including Mexico, which in reality is in North America, but people are ignorant of geography). And we assume he will speak Spanish (or Portoguese, but for us they sound quite similar).
So the distinction is not really geographical, it is more language-based.
We think that North American means speaking English (albeit in Canada there is also a lot of French speakers). And South American means speaking Spanish.