"United Kingdom of Great Britain and Northern Ireland hasn't been the formal term for centuries"

Of course it hasn't and no-one said it had been. What I said was the formal term for the state was the UK and had been for several centuries. Prior to the 1920s, and since 1801, the full title was the United Kingdom of Great Britain and Ireland. The only difference since the 1920s has been the change affecting Ireland. The term is still the UK though - as it ws prior to the 1920s.

Of course the collective term for the islands isn't the UK and Republic of Ireland. That is simply a 'partial' list of the political entities that constitute the British Isles. As I said the term British Isles long predates the names UK, Ireland, England, Scotland, Man or whatever. It is the only collective term that would be recognised by the vast majority of people in these islands. I have seen other suggetions one example being the "Atlantic Islands" but really if you used that virtually no-one would know what on earth was being talked about.