I think there is a European culture. We share a common history, basically going back to the Roman Empire and then, later, two principal strands of Christianity (Catholic and Orthodox) from which a large part of our common heritage (architecture, music, literature, aspects of politics and much more) derives. The various empires and ruling houses that have come and gone make us all interconnected in many ways. Our artists and scholars have often transcended national boundaries and so has trade. Many of us think of ourselves as Europeans and for many, certainly me, it was a source of pride to reflect on our common access to this rich heritage.
Every person is in fact a member of numerous overlapping identity categories. There is nothing mutually exclusive about being British versus being European. Like many of my countrymen, I have both lived for a while in another European country (The Netherlands), and married someone from another, different one (France). In fact, I find it hard to forgive my country of birth for snatching away the European citizenship I valued so much. This is a not uncommon feeling.