The Isle of Man isn't a part of Britain though. It isn't even a part of the UK. It is like the Channel Islands a Crown Dependency.

I understand that for historical reasons many Irish people don't like the term British Isles because it seems to associate them with Britain and almost sounds like the islands belong to Britain. It is though in truth purely a geographic term and in truth it comes from a term which predates even when the term Britain became more specifically associated with the larger island.

It is kind of spitting in the wind as the term is so old and ingrained. Though the UK gvt did take some note of the terms when in the 70s it laid down officially what the area associated with the UK would be called. That is the UK itself plus the Crown Dependencies are officially known as the British Islands which is different from British Isles which is a geographic term for the whole island group. Though I think hardly anyone actually uses the term British Islands. People tend to not think much about the Crown Dependencies so just ue the terms Britain or UK for it.

You can't of course legislate for how individuals view themselves anyway. Plus their self definition changes. I've always viewed myself as Scottish though I have no issue with British as a secondary thing - but more in a geographic sense. If we were independent I'd still view myself as British in a geographic, cultural sense - like Swedes etc are Scandinavian. My wife though no longer views her self as English - which she used to do. She now only identifies as British. She is very pro-union and this change has only happened in the last few years. We are all different. I think by viewing herself as only British she is pushing aside the idea that Scots and English are somehow different.