'British' is [also] the adjective for people from the United Kingdom (UK). The full name of the UK is the 'United Kingdom of Great Britain and Northern Ireland'.
Folks in GB can generally distinguish someone with a Northern Irish accent from someone with a more southerly Irish accent.
e.g. Belfast, or Derry accent vs Dublin accent.
Not surprisingly so since Donegal, Cavan and Monaghan were also extensively settled by Scots immigrants, but were cut off from what's now casually called Ulster in 1922. So they had 300+ years of shared language versus 100 of separation.