I was born in the UK but with a West Indian and an actual Indian grandparent on one side. Lived there my entire life up until relatively recently. There were still people who would consider me not British. When people either in the UK or where I live now ask where I'm from and I tell them I'm British, there are many people who say "but where are you really from?"
But it doesn't mean anything if I say I'm Jamaican, or Indian, because one I've been to for some odd trips as a kid and the other I only transferred through an airport. Yeah my DNA shows that, but my entire life has basically minimal connection to either of those places and a continuous connection to Britain.
I got the accent and the passport, but I didn't get the skin colour. So these people will always exist who want to make it seem like I'm not "really" British. But that is on them, not me. I am British, whether they like it or not.
This is really a long winded way of saying: there will always be some people who consider you not truly British. Fuck them. They are idiots that have at best shackled themselves to some outdated view of what it means to be "British" and at worst want to shutter the whole country off to anyone who doesn't look or speak like them and pull us all back to the stone age.
I think if you speak to people you are close with about this they would consider you British. If you speak with Baz down at the Red Lion he'll ask you for a pierogi and then fall asleep in his own vomit after a few too many pints of carling. I think Baz is much less important than all of the real people in your life and most of all, the way you view yourself.