Many of the people I chat to from Europe on IRC confirm that notion. They consider themselves E.U. citizens first and foremost.
Of course these are heavily tech-oriented chat rooms so I'm not sure how far that can be applied. But I find it interesting myself, especially given the history of the USA, where right up to the 1870s people considered themselves a citizen of the state first and not the nation... obviously it has eventually inverted.
But should this inversion actually happen, when does the USA get to ask for the E.U. to act as a singular entity for things like Olympics, U.N. representation, and other international protocols? ;)