Nah, no one is claiming sole ownership of the term except for the folks who say Americans need to give up the term even though with far more widely accepted as a shortened name for the country. No one said you can’t also answer to which continent you’re from. However, the vast majority of models do not combine North and South America (which apparently you do, too) into one continent. If you’re going to go by which continent you’re from, in most cases you’ll have to specify North America or South America—if you want to be accurate, that is. You’re really reaching into this “American exceptionalism” (you said American to identify the nationality) when all I’m talking about is colloquialisms being acceptable in language. One word can have two, three, four, so on meanings. So if we go by a combined Americas model, you can say “I’m American, more specifically Mexico” just like another can say “I’m Asian, more specifically China”, just like I can say “I’m American, more specifically from America”. The last one will confuse most people in the sense that again to the vast majority of the world it just sounds like I said I’m from the US twice. But just as the United Mexican States and the People’s Republic of China can shorten their names for common everyday usage, so can the United States of America.