I have never heard anybody, anywhere, ever, refer to somebody from the North or South American continents, who is not a US citizen as an "American". "The Americas" commonly refers to both continents, but I've never heard anybody call a non US-citizen "American". Where is this a common usage?

From what I've found its really only from predominantly Spanish speaking university students in South America.

It also ignores most colloquial usage of the terms America and American in English. While technically correct, it is unlikely to occur. Americans are citizens of the United States of America. Just as Mexicans are citizens of Estados Unidos Mexicanos. But we shorten them to Mexicans as well.

Its more useful in either case to just list your nationality in either case. Unless people seem to want to identify with an Italian name for both continents I don't see where getting all up in arms over Americans referring to a citizen of the US. USian is an alternative, but lets be honest, it rolls off the tongue horridly and is unlikely to catch on.

This mostly tends to come from people who have nationalistic quibbles, and who want to propose alternate "I will force a name onto you that expresses my attitude toward your country" schemes.

Also, it's an unsolvable problem, because there is no part of "United States of America" that is technically unique. It's not the only nation on the American continents, and it's not the only nation whose official name includes "United States" (Mexico does as well).

Technically correct, but I guess it depends on context, both cultural and conversational. In the UK for example, "America" and "American" always refer to US not the continent, unless there's specific context surrounding it to suggest otherwise. I believe that's generally the case in the US as well, and I suspect (but could be wrong) in most countries?

I think it's technically incorrect, actually. In standard English, American is the demonym used to refer to citizens of the USA.

People seem to make a number of wrong assumptions on that, including 1) it can't also be the demonym for anyone else and 2) other countries can decide another countries demonym in their native language. Both of these things are patently false. If someone wants to use American to refer to a member of the Americas (continents) then they certainly can, but they should include enough context to disambiguate. Moreover, other nations don't decide what a nation gets to call itself in its native language. We call ourselves Americans -- so we are Americans. If other countries don't like that, well, tough for them.

This only applies for countries in South America and maybe, maybe Spain but I really doubt it.

Go anywhere and the planet and tell people you're 'American' and not one of them will think you meant Brazil, or Canada, or anything other than the United States.