As is Scotland. Try asking the Scots if they’re British. I think you will receive the same response in Wales. Being part of the UK is certainly true, but it doesn’t mean they’re British. The “United” part of the “Kingdom” is only so they won’t continue to bludgeon each other into obscurity.
I work with a guy from Wales - he has a thick Welsh brougue. About once a week, some makes the mistake of calling him English. The response is usually ugly, but (quietly) amusing for the rest of us in the office.
Occasionally, someone will mistake him as Australian. He doesn't go so ballistic over that, for some reason.
The UK is England, Scotland, Wales and N.Ireland
GB is England, Scotland and Wales. Hence British.
Historically speaking, the Welsh are the only true “British.” They’re descended from the original “British” who were conquered by the Romans.
The English came in after the Romans left (wise decision on their part) and took over most of the island, forcing the original British back into Wales, Cornwall and even across the sea into Brittany. The English (and everybody else on the islands) were eventually conquered by Normans from France.
The Scots, at least by that name, were originally invaders from Ireland into the northern part of GB.
Of course, all of this is largely genetically meaningless since the various original stocks have been enthusiastically cross-breeding for a couple millenia.