Things that people used to call 'typically French/German/Italian/Spanish/Slovak/Greek' are increasingly seen as typically European. I can't back this statistically, but virtually all the people I met in my life who have lived in at least two EU countries think this way. I honestly can't think of a single person who I've known and thinks otherwise and I should know because this topic almost always comes in discussions when you get to know somebody.
Of course, there are some minor cultural differences, but these are not particularly larger than the ones found in the US, India, China, Brazil, or the Russian Federation for that matter.
The difference in culture between the UK and other Northern European countries is often exagurated. Many say that the UK is more like the US. In my experience this really isn't true.
Have you actually travelled in Europe? Germany is radically different from France; both are radically different from the UK, and let's not even start to talk about Eastern Europe.
The degree of deference to authority, the specifics of the rules of the road, the level of interference of governance - these things vary, but they're within a spectrum. Things go off the wall different outside Europe.
I think the problem with EU is primarily one of communication. The member states should do more to inform the public of what is going on in the EU, and EU should do more to avoid negotiations behind closed doors (because how can you ever trust a process like that?).
No, it's not mutually exclusive but the order matters. IMO American first see themselves as American while here a French guy will see him as French and then maybe European. That's the sentiment I'm referring to.