Anyone can become British, but not everyone can become Scottish, Irish, Welsh, or English or even Cornish come to that.
It comes down as much as to whether other people perceive you as such.
Racist or not, people tend to mean white indigenous people because being Cornish or English etc is a more personal thing than just being British.
It's fairly universal IMO.
Even if I was born in India a lot of Indians wouldn't see me as an Indian. I could wear a sari, but I'd look different, customs or no.
Same with China or Japan.
I wonder how long it would take to be fully regarded as Japanese?
I don't think I ever would, no matter what my passport or the law says.
Being English or Scottish also means sharing a perceived culture but that does vary.
It is racist because it has tribal roots.
If you were black or brown you would have trouble passing off as Cornish.
It's not you can't claim it, or that the culture and accent isn't there, it's the majority won't accept you.
However somewhere like Birmingham which has large numbers of foreign different ethnic communities, you might get away with saying you were a " Brummi " however
In Somerset, Devon and Cornwall you wouldn't totally be accepted as a local white indigenous person would be.
Sometimes Cornish people take the piss out of people from Bristol and even in Somerset among the older generation anyone north of Bristol is considered a foreigner
Birmingham is " up north"
It's not just my perception of Somerset though, I was once teased by a Yorkshireman while in Yorkshire for being a foreigner because I came from Somerset.
I think he had encountered the old Somerset attitude ( or old Yorkshiremen have the same attitude to those in the south)
Basically being accepted into personalised groups is as much about others accepting you, as what you claim.
English is a bit wider than say Cornish or from Somerset, but there is a general perception of what someone is like and if your face doesn't fit, people won't see you as Cornish.
I think it's the same in Some parts of Wales.
Plus in some places it's not just about where you are born but where the generations of your family were based.
That's another aspect.
I think lots of ethnic groups in the uk retain their own personal identity on top of their own British one, because they recognise the difficulty of being recognised as say Welsh.
So people are British and Pakistani at the same time.