Actually European would be a persons nationality and French would be their ethnicity. Just as Asian is a nationality and Indian, chinese, ect would is the ethnicity, but anyway:
It just is wrong. All people from Asia aren't considered Asian.(According to the us census) In India they don't even recognize race, it's all about a persons nationality/ethnicity.
We as Americans want to make people fit in our social construct so we assign arbitrary races to people that make no sense whatsoever. In realty the idea stems from black and white people, and was extended to include others using a mix of ethnicities and nationalities to classify people as a "race" races have changed over the years. The categories for race have changed multiple times on the us census. At one time it was just black, white, Latin now it includes over a dozen different selections. Race is totaly a social idea and as nothing to do with where a person is from.