You can be both and how you identify is important.
I was born in another country and was there until I was 22. Then I moved here, on my own.
I will always identify and sound like I'm from there, it's part of how people see me and view me, and how I view myself. Thankfully for me it's a nationality that is generally popular or viewed of affectionately by most Brits, and I felt embraced into British society quite quickly.
I've now been here more than 25 years. Everything "adult" I've done, like learn to drive, buy a house, have kids etc has been done here. My accent has softened considerably. When I go "home" they think I'm from here. I'm a bit no man's land in that regard.
I sort of feel British as well. Apart from having the passport, I also have the blood, and am a pretty equal split of English Irish and Scottish (did a DNA test). If someone told me I wasn't British, I'd have a strong argument to disagree, as nearly as strong if someone back home told me I wasn't that nationality.
I wasn't born here though.
None of it would upset me I don't think, as how I feel is more important than how others view me.
I think in my situation being a native English speaker and white probably makes things less complicated as well.