Basically that's because "American" isn't a breed, like Scots or French. It's a political choice and an identity which is (should be) open to anyone who lives here and adopts the Constitution as the set of laws they want to live by.
Ironically, this is why Yarvin and Musk and Thiel will fail in their attempts at overthrowing the US government to create "network states". The US is the original state founded on a set of ideas rather than identities. (Since, at least, the Roman Republic). That is its strength, and something difficult for Europeans to really understand.
For that matter, the UK, despite being a country, has some aspects of this; while Northern Ireland in particular is complicated for historical reasons, plenty of English and Scottish and Welsh people would think of themselves as English or Scottish or Welsh AND British.
This is a very common conflict for many communities around the world and rarely is “just be both” a satisfactory solution: https://en.m.wikipedia.org/wiki/Separatism