They might try to give that impression after they arrive here illegally, but the fact is they already a name. If they are from Mexico, they are Mexicans. If they don't like who they are, they shouldn't try to steal our name because we only have the one.
We are Americans. As in United States of America.
The folks I was referring to are middle to upper class Mexican nationals, with no intent of ever entering the US illegally. They do come here frequently on business though, and one couple and their children currently live legally in the US, managing the US end of the family business. The kids are actually American citizens, having been born here some years before when their grandfather was running the US end of things. He's now back in Mexico, running one part of the business there.
They do of course generally call themselves Mexicans, but on occasion they say they are Americans as well.
I wouldn't get too hung up on the terminology anyway. They know the difference between USA citizens and EUM Citizens. (Official name of Mexico is Estados Unidos Mexicanos (United Mexican States).