Well, he's not wholly wrong as Ireland is the name of the island as well, and he makes some indication that United Kingdom is technically a theocracy (which would be wrong), but the gist of what he's saying is about right. The part about Northern Ireland being British not Irish I believe is in respect to the passport/citizenship issue. Although it's pretty common to hold both the default in Northern Ireland is British, not Irish, citizenship.
Very much agree until that last two points. British, as an identity, didn't kick off until the early 1700s. Britons (or Brytons, if you prefer) are a different kettle of fish and refer to the people on the island of Great Britain (south of Scotland I think?) pre-500 AD or so. There's no link between the two except in approximate spelling.
To be honest, I'd prefer if they did Irish, Welsh, Scottish and English civs and then kept the Celts more focused on their European incarnation (a la Civ 4), but I'm a bit biased.