There is a certain European identity for sure. But those who subscribe to international identities usually just go global. And the nationalists, while they consider "being European" an important part of their identity, for them the nation comes first.
>unified Europe
No.
EU's free trade is important to keep peace in Europe, and international relations - from trade to such communications we have here on this board - help to understand each other. But the real reason why there was no war in Europe, why the European state decided to cooperate with each other, is that the USA kept a defensive military umbrella over Europe, keeping it safe from outside attacks, so noone had real need to militarize, keeping defense spending low and more for economical development. In other words the USA keeps all European countries under military occupation preventing them from fighting each other. In other words the USA tells her European client regimes not to fight with each other. In other words mommy USA holds her kids by their necks and doesn't allow them the punch each other in the face.
If the USE bugged out of Europe each country would have to rearm themselves, creating the security dilemma of Realpolitik. Sure we are all friends here, a big family, right. But defense doesn't calculate with rhetoric, but potential. If France started to arm herself, doesn't matter what was the official explanation, Germany the next day would eye her suspiciously, and saw they could take fugging Baden, and Germany would start arm herself too to keep pace. Which would make Polan real paranoid for sure... etc.