Many of them identify as "African American", "Italian American", etc. They state their ethnicity first, and American second. But of course, they are joined as a nation.
The EU is a different concept in many ways. In the EU you're trying to legislate for fully industrialized nations, with different traditions and languages. The US has the advantage of having come together during their formative years. The EU isn't like that. If the US were coming together today, I'm not sure it would work the same.
That said - the EU being the way it is is a good thing, imo. The French do not want to speak another language (unless it's a second language). They don't want to be known as citizens of the EU, they love their country, and they want that to continue. The same is true in the UK. How would the British feel if German became the common language? It just wouldn't happen. Just look at the hatred immigrants get today - we don't even want foreigners coming here!
On the other hand, it's the diversity that makes the EU such an interesting place. Within the EU each country is fighting the goals of their own citizens, which isn't any different than what happens in Congress.