> The British Isles are a group of islands in the North Atlantic off the north-western coast of continental Europe, consisting of the islands of Great Britain, Ireland, the Isle of Man, the Hebrides and over six thousand smaller islands. [They] include two sovereign states, the Republic of Ireland and the United Kingdom of Great Britain and Northern Ireland.
No-one in Ireland would refer to them as the British Isles.
Simply Britain and Ireland (or Ireland and Britain) is far more commonly used if referring to the archipelago.
The talk pages and edit history for the Wikipedia article on the British Isles term are interesting. You can see the arguments and edit wars. Much like the insistence on naming the country page for Ireland the "Republic of Ireland' (which is our football team and country descriptor, not the name of the country), I think Wikipedia are on the wrong side of history here, but some editors até vociferous in their objections to change.
It's not a British Isle -- it's an Irish one. Great Britain refers to the main island composing England, Scotland and Wales. The Irish isle of Ireland is composed of the two Irelands (Ireland and Northern Ireland). The 'British Isles' term is disputed by many Irish people. The reason it's not a geographic term is because it's not a river -- Britain is a geopolitical entity not a purely geographic one.
The fact the Ireland was head of the GDPR for the EU was merely an odd coincidence and has been rectified. You won't many upset Irish people over this.