Being Northern Irish, he is eligable for either a UK Passport, or ROI Passport. Even if he chose a UK Passport, it'd say he is a Citizen of the 'United Kingdom of Great Britain and Northern Ireland'. The UK doesn't equal Britain thus he is not British.
the OP is right though. Look at the size of this; please log in to view this image compared to this; please log in to view this image The bottom one is MUCH bigger
Britain is England, Scotland and Wales. The UK is Britain plus Northern Ireland I learned that in Primary School
The Americans never quite get it right - They had the decency to murder John lennon but then they decided to immortalise him in Central park rarther than flush his ashes down the khazi and tell everyone that he was an idiot who wouldn't know a decent tune if it moved into his house - idiots.