The people who care about Europe and European culture, and who identify as Europeans, tend to think of the current European Union as an anti-European institution. At least that's what I've noticed in recent years.
Ah, I see. It's interesting. One would think that a monetary union would be a component of a unified Europe in the sense of reducing friction and improving economic exchange. However, it cannot be logically applied without the political machinery that requires unification in a more rigid sense. Money is a ledger of credits and debts, but to date has been inextricably linked to the levers of political power. Would the EU be stronger without the Euro? Maybe. The inconvenience of exchange might be better than a monetary union that must override democratic governance that is intrinsic to cultural and national identities.