Think of it this way, if you're from New York you're a New Yorker, an American and a North American (in the Anglophone definition anyway). In the same way French citizens are European citizen. It is very important to note the European Union isn't a country (it's a union of countries). Yes, you can become a citizen of the European Union but to become a European citizen you need to have the citizenship of a member-state of the European Union (EU).
It's not a stupid question at all.
The nationality of French people is French because France is a sovereign country.
However, there is an European nationality and everyone from a country of the European Union has the European nationality.
So if you have the French nationality, you automatically have the European nationality.
But most of the people don't even know they have the European nationality because it has really very few use.
French
French.
And there's no such nationality as 'European', but you can say someone is European. It is not, however, a nationality, merely a description.
Your nationality is French but since France is a member country of the European Union you also hold the European Union citizenship which gives you the right to live and work in all countries of the EU.
I am kind of surprised some people are considering the EU a country....it's just some pseudo-economic union, not a real country.
You are from France = you are French.
You are both. I'm a Missourian, and an American by birth.