No, really, I'm American

There's a discussion I've had from time to time. It goes something like:

Interlocutor: "What nationality are you?"

Sasquatch: "I'm American."

Interlocutor: "Well, I'm Mexican, which means I'm American, too. You should call yourself 'a United States citizen.'"

Sasquatch: "Well, I'm from the United States of America, so I'm American. The official name of Mexico is Los Estados Unidos de México, but you don't seem to have a problem with calling yourselves Mexican. So why shouldn't I be allowed to call myself American for my nationality. What gives?"

The conversation then devolves into an "am too/are not" and ultimately goes nowhere. In Spanish, the descriptor for someone from the United States is estadounidense, essentially "unitedstatesian." If someone is from the United States of Mexico, then why isn't he/she estadounidense as well?

If you're native Spanish speaker who can shed some light on this seeming inconsistency in the application of estadounidense, I'd love to learn more about it. If you don't speak Spanish and have a theory that might hold some water, I'd love to hear that, too.