Culturally we generally consider Europe to mean continental Europe i.e. Europe without the islands. Thus the U.K. being an island nation isn’t part of Europe.
But the definition of Europe will change depending on who you’re talking to and what your talking about.
In my experience, the UK is culturally very similar to the rest of Europe (much more similar then we are to the US, although there is some influence there). This fact is just obscured by the language barrier.
Mostly British people's poor education in languages. It's not that other European languages would be hard for us to learn, it's that most people in the UK de facto don't learn another language to a useful level (presumably because English is so widely spoken that we can get away without doing so). Which makes it very difficult for most of us to engage with European culture, media, etc even though it's often quite similar to our own and I suspect would otherwise interest us.
But the definition of Europe will change depending on who you’re talking to and what your talking about.