Calling yourself Americans, after the entire dual continent

But we never use "America" to refer to North and South America collectively. You can say "the Americas", or just "North and South America". And there's no adjective that means "of the Americas"; you can say "North or South American". But just "American" unambiguously means "of the USA".

I've always wondered if disagreement over this comes from the fact that in some parts of the world, North and South America are considered to be one continent just called "America", whereas we consider them to be two separate continents. And we don't have a word for the pair of continents, any more than we have a word for Europe and Africa together. (I mean we do have "Eurasia", which kind of pokes a hole in the hypothesis, but maybe that's a special case because a brief glance at a map makes it clear it's pure fantasy to count those as separate continents.)