The use of "American" to designate residents of the United States dates back to a time when there were no other countries on the continent. If you lived on the American continent in the 1700s, you were either Indian, or an English, French, or Spanish colonist (or a slave). In the 1770s, 13 of these colonies rebelled and formed their own union. It wasn't really a country though, since each had its own money, laws, and government. To differentiate them from colonial citizens, Europeans called them Americans. Most Americans thought of themselves as citizens of their state (Virginians, New Yorkers, etc.), and in fact it wasn't until after the Civil War that they began referring to the United States as a singular entity, rather than a plural.

Your description of America as a single continent with "plenty of different countries" betrays a very Afro-Eurasia-centric view of the western hemisphere. Unless you can point out Wisonson on a map, or tell me what the Federal Republic of Central America was, maybe keep your eastern hemisphere judgements to yourself.