Yes -- and what ethnic terminology such a person would prefer to use is a complicated and delicate issue, and I think you should call them whatever they want to be called (within reason).
But I certainly wouldn't suggest listening to the opinions of random Anglos who aren't affected by the issue at all claiming that "calling someone Mexican is offensive"
Perhaps 'actually Mexican' in this context means Mexican ancestry. For communities struggling to be accepted as American, this is a legitimate issue.