It's a long lost battle, but whenever I hear somebody described as 'American' I still think to myself 'you mean, citizen of the US, right? Not just from that continent'. Those would be 'American' whilst I would 'only' be 'South-American'. It shouldn't matter, the name only refers to an Italian cartographer and was brought about after Columbus' third voyage when they 'discovered' the mainland which, incidentally, was the Western coast of Venezuela around Lake Maracaibo. There's always seemed to be a little bit of stretchy meaning to the word. We consider ourselves American (as in the continent) and older people, particularly in the south of the continent, would bristle and proclaim 'wait, _we_ are Americans'. There was always the ironic mention of the Monroe doctrine of 'America for the Americans' which we used to read, in our corners of the world, as 'America' meaning the continent and 'Americans' meaning US citizens -not us. Episodes like the Chilean 11 of September (1973) didn't help.
I s'pose somebody could come up with the American (the continent) equivalent of 'Azania', a word that related more to the peoples that inhabited it before the Europeans went in. But it would never catch on and... it wouldn't apply to most of us in any event. Most of us down there are mongrels, the product of the atrocities that the Spanish, in particularly, committed, which resulted in a mix of 'races' (there's another slippery concept) European, indigenous and black. That I think is actually a good thing.
Maybe I should stop replying with things like 'You're American? ah, I'm American too. My part of the continent was America before yours was'. There's no point. It's a long lost battle. It would only antagonise people who often are warm, lovely peeps (at least the ones I know here; we'll leave Trump supporters for another discussion).