Yes I get what you are saying, there's a different ambience around the notion of cultural identity. Perhaps that lack of ground is what I have perceived as a weird hollow (lack) at the centre of American life when I have visited. I also felt that as opportunity though.
It's not as if everyone just arrived there though. And how much of modern British cultural identity is really built on stuff that's older then the USA? Obviously there are parts of the UK where people can agree on what it means to be English, Scottish or Welsh and feel themselves to be that, but I think it's quite different in the large metropolitan centres and London in particular.
I've never been to the U.K., but every other European country I've visited felt worlds away from the U.S. culturally, some of them better some worse.
The first thing I noticed was that in Europe people obey laws. It doesn't matter which laws, or how trivial they seem, people do what they're supposed to, even when it comes to things like jaywalking! In Germany this was especially striking--they didn't just obey laws, they seemed to have a strict social code of conduct that was adhered to without question at all times. For example, I would simply cross the street when there were no cars at whatever point I liked and people looked at me like I was just out of my mind. Don't talk to Turks. Never ask "how are you?" unless you really mean it. In the U.S. nobody obeys laws, no matter how important they are, unless the fear of recrimination is tangible and the punishment severe. Law and law enforcement are there to protect you from harm, not keep you from doing whatever you want to do.
The second thing I noticed in Europe was that people all eat meals at regular times, and with their families. I don't really know anyone who does this in the U.S. except my ultra WASPy boyfriend's family. (well, I don't know, they're German and his dad's Jewish, but they act WASPy) And when they do it's awkward and there's lots of forced conversation until everyone gets sufficiently drunk to deal with each other. Also, Europeans didn't snack or eat nearly as much junk food as Americans. Europeans on average are much thinner than Americans, which is nice.
The third thing I noticed was that people in Europe take their own religion (or in England's case, I imagine, social etiquette, which is sort of like its substitute for religion since historically no one took the Church of England seriously) seriously but not as a set of beliefs, more as cultural tradition. This is very refreshing to me. It's fun to look at pretty cathedrals, not so fun to listen to fundamentalists thump the Bible.
The fourth thing I noticed was that European people all assume that others are nice, good people for the most part, or at very least respect others. Neighbors are tolerated not sued. People know their neighbors or are at least friendly with them when they see them. Once I left my wallet in a cab, with lots of money and an ID, and I thought for sure it was gone forever. Someone else called the cab authority for me and the guy brought the wallet back full of money! I tried to give him $50 for his trouble and he wouldn't take it, he couldn't understand why I'd want to give him money. In the U.S. not only would that have been stolen, but my identity would have probably been thieved, too. Here people assume the worst about everyone.
There are a lot more but those are the ones that struck me the most. They sound pretty minor but they add up to huge differences.