This again? I think your point is appropriate when speaking Spanish, Portuguese (EDIT: not Portuguese), etc. However, in common English (no, not just in the US) the word "America" more often refers to the USA. Languages don't use the same or derivative words for many other countries as well.
Not even in Portuguese. "Americano" specifically means "someone from the United States". Spanish is the only major European language in which its local equivalent of "America/American" does not refer specifically to the USA.
That's bad and wrong, you should think past your familiarity for a minute and logically not do it.
This shouldn't be mixed up with the other common case which is often discussed, 'American' as the demonym for people and things from the USA. That isn't quite as bad (though it still annoys other people from elsewhere in the Americas) because it's using the name of the superset - Americans, from the USA, are also Americans, from the continent. Whereas, when "America" (the USA) does something, it doesn't necessarily mean all of America does it. The USA is a subset of America, not a superset or the only entry.
Language isn't always logical, but if you won't draw the line at something so clearly backwards as using the whole continent's name for the actions of one country, you can only expect it will start an argument every time.
I disagree with the premise here that "America" describes the continents of North and South America. That is in and of itself the issue here, so I don't agree that the USA is a subset of America because I would refer to the continents as "the Americas" or similar.
We chose a really difficult country name to create an identity around. United States of America citizen just doesn't have the same ring to it as American does. Meanwhile, you know, we also stole Amazon as a word to mean something totally different than a river and region in Brazil.
wut... amazon is a greek word, which is actually derived from a different older language (the hypothetical pelasgian, the original indigenous people in the balkans).
So, yeah... you can say brazil borrowed that word from the greek language via latin, and the greeks themselves borrowed it from an extinct language...
Am·a·zon
/ˈaməˌzän,ˈaməzən/
Learn to pronounce
Origin
late Middle English: via Latin from Greek Amazōn, explained by the Greeks as ‘without a breast’ (as if from a- ‘without’ + mazos ‘breast’), referring to the fable that the Amazons cut off the right breast so as not to interfere with the use of a bow, but probably a folk etymology of an unknown foreign word.