It’s a weird way to phrase it. I actually have dual citizenship. I say “I’m X and Y” not “I’m X or Y”.

By the way, you’re technically correct when saying "American or European " because you come from the Americas (the continent). But if you’re referring to the United States of America nationality, then you’d have to say the other country’s nationality as well.

But back to the main point, let’s say that you’re about to become French. You’d say "I’m American and French. Or “French-Americam” or “American-French”. “I’m French or American” sounds weird.