TL;DR accepting intellectually that we are technically Europeans has very little to do with feeling like European-ness is a part of one's identity.
I thought it'd be obvious that I didn't mean every Irish person would be bemused, but I should have been more explicit.
There's a difference between accepting Ireland as legally and bureaucratically part of Europe, or thinking it's politically or culturally useful to be in the E.U., and feeling that European-ness - however you choose to define it - is a part of our identity as a people.
My anecdotal experience (have lived in two European countries and one non-European country, learning the local languages and diving deep) is that European countries are about as "foreign" to Irish people as any other countries. There are exceptions, and it depends on the country, e.g., we're closer to the Poles now, and have a few fun stories about crates of Prazsky at the wedding in Gdansk.
Try out the experiment - ask an Irish friend or relation of yours if they feel like them and a Finnish / Belgian / Romanian person have something in common, or what they know about that country. My experience is that a lot of people (not everyone, and again, anecdotally) are very uncomfortable even with the idea of imagining that other places really exist.