At least for Scots, this is not a general case. Some consider themselves not to be British because they don't want Scotland to be part of the UK, others will take exception to the conflation of "British" with "English" because that implies that Scotland is just considered part of England. You don't even have to have strong feelings either way about either England or the UK for that one.
At least for now, the word "British" is associated more with the political entity of the UK than the geographical entity of the island of Great Britain. That most of Scotland is on the island of Great Britain will not persuade anyone in the first camp.