I've never met an American who said they were British/Irish/French/Italian/whatever and meant that they were literally from that country the same way as someone who was actually born and raised there, and, being American, I have had a lot of conversations with Americans about genealogy! I think both Europeans and Americans tend not to realize that the other means something different when they use these terms.
One thing that also rarely gets mentioned in these conversations is that it's not actually that common for Americans to claim a single non-American heritage either. It's usually more like, "I'm German and Polish on my mother's side and French and Swedish on my dad's side."