I don't think anyone is denying them the right to call themselves American if they so choose, but as this whole conversation illustrates, the term is incredibly ambiguous. When the argument is that "American" could mean anyone from the Americas, that effectively covers the entire western hemisphere which is a bit nonsensical to me. The point they're trying to make is exactly the problem. There are lots of Americas.

Frankly, I don't understand why this is so much of an issue. There's no continental culture and solidarity connecting the northern tips of Canada to the southernmost parts of Chile. Nobody is trying to erase a deep historical culture of America being one group of people. Why are these people trying so hard to create a continental identity that doesn't culturally exist? I honestly think the point is to take the term "American" away from US Americans just because.

I've never heard of a person from Peru or Brazil or Guatemala deeply yearning to call themselves "American" meaning somewhere in North or South America, but refraining from doing so because they feel marginalized. Feeling marginalized isn't why they don't do it, they don't do it because saying "I'm American" meaning continents is useless. You might as well say "I'm from somewhere on the west side of the Atlantic." The term "American" becomes pointless if you mean it that broadly. Imagine someone from the other side of the Atlantic saying "I'm Eurasiafrican". There's no culture that connects all of those peoples, it would just be a pointless moniker.

Cool, but my Korean phone wasn't designed by a company based in Pyongyang. And go ahead and ask a Canadian how they feel about being called "American".

North American, Central American, and South American are all terms you can use similarly to European, Asian, or African.