Yes, I've spent time in Northern Ireland. And I was objecting to the generalisation that suggested that seemed to suggest that everyone on the Protestant/Unionist side of the divide didn't consider themselves Irish. And I note that GUEST (you must have a handle of some kind) agrees with me on that.
But even where people see themselves as "British", in most cases that doesn't mean they don't see themselves as Irish as well. No one would suggest that a Scot who doesn't support independence for Scotland isn't a Scot. Even if they might think he or she was a fool.
And some of the most extreme "Loyalists" don't even in any sense aspire to be British, but something totally separate.