Ireland is the same. Being an island, like Britain, we are geographically cut off from (continental) Europe, but also somewhat culturally. How many of us on these islands would say "we are European"? Not many I'd say. We are European of course, but it's not something that we would put down as our identity. Europe (continental) feels connected. So many are multilingual, we very much are not.
Even in football, our teams (my own being Liverpool) who qualify for UEFA competitions, "play in Europe", as though it's some separate entity.
In Europe:
Drive on the left? Only us.
Use miles, yards, feet, inches? Only us.
Use common law? Only us.
Not being as connected though doesn't make here boring. It's probably the opposite as who wants the same?
The UK is a cultural superpower, being a global leader in literature, music, theatre, sport (oldest golf tournament in the world (the Open), oldest tennis tournament in the world (Wimbledon), oldest football club in the world, the most famous sports stadium in the world (Wembley)), television (BBC the world's oldest national broadcaster), comedy (British/Irish comedy cannot be surpassed), film (first moving picture filmed in Leeds; Roundhay Garden scene), the birthplace of Halloween, of modern Christmas customs (Dickens), etc etc.
There is also history wherever you walk (in London the blue plaques on buildings commemorating famous people and events). Over 1000 years of history, from castles in the 11th century to the world's first iron bridge in the late 18th century (an iconic symbol of the Industrial Revolution).
https://en.wikipedia.org/wiki/The_Iron_Bridge