I’m not sure this is correct; in general many people would think of themselves as X and European, where X is the country of origin or citizenship. The EU is not a country.

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.

America is a bit different and, for some generations, consider ourselves better than the ugly factionalism and national identitarianism that led Europe into centuries of pointless warfare. One can be an African-Mexican-Asian-Queer-Irish-Jewish-Italian-American without subjugating one's identity.

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.