After all, lots of countries have political subdivisions called "states". A bunch even advertise the fact that they're made up of states in their official name. Some even go so far as to throw the word "united" in there. So in some bizarro-universe, the abbreviation "U.S." or colloquialism "the States" could refer equally well, to say, the United Mexican States as it does to the United States of America.
But Canadians would never refer to their other southern, NAFTA-loving neighbor as "The States" or the "U.S.". Every sane person just calls it "Mexico". Of course, the continents aren't named North and South Mexico. But maybe if we all had goatees... All the other cool countries get to remove "Federated States of", "Grand Duchy of", "Republic of", etc. from their common names. Why can't the USA?