This is complete nonsense, especially the bolded. The "Celtic Church was separate" is a completely discredited idea that's frankly persisted as an AH-meme. The so-called "Celtic Church" was little more than a different set of rites and practices unique to Gaels, fully in communion with Rome.
Seriously - do we have to have a discussion about persecuted religious minorities being little more than nationalist contrarians?
Furthermore it utterly denies the role Ireland played in spreading Christianity to England and other continental Germanic lands
Even then, "We're both Catholic, just like most of the rest of Europe" is less of a uniter than "We're both Protestant, unlike the filthy Papists on the Continent who want to take us over and convert us to their filthy Papist ways."
Excuse me? Was such language needed to bring your point across? Besides, when you take religion out of the equation, wasn't that what England and later Britain was doing to Ireland? Sure religion might have developed into something to defy the authorities. Heck the Generality Lands remained firmly Catholic under the domination by the Protestant Dutch Republic (having no regional assembly nor representation in the Estates General didn't help either). OTOH why is it hard to believe that people stay Catholic out of their own conviction?
England, certain German states and Scandinavian countries in part became Protestant, because their monarchs wanted to obtain the material possessions of the Catholic Church (lands, monasteries). I agree, that is not the whole picture, but that part shouldn't be ignored either.
As for the uniter part neither Catholic nor Protestant countries were ever hindered to start a conflict with a regional rival, because they had the same religion.
This leads to my conclusion, that whether Ireland would be Protestant or Catholic, if the rest of their treatment is mostly the same, I'm sceptical they will love England/UK more than they did IOTL.
@VVD0D95: shared language, yes today, but Ireland had and has a language of its' own. Europe is filled with regional languages of groups, which all feel a big centralizing power tried to exterminate their language and culture. To some degree, Ireland might even fit in this group too, at least with regard to the mixed relation with the former oppressor.
These things tend to last very long, just look at Belgium, where effectually the tables were turned with Flanders now being de facto dominant over the Francophone part of Belgium. Yet both parts to a degree are still stuck in the old roles. Granted the Flemish elites used to talk French as well, and the Belgian Francophone commoner was just as bad off as his Flemish counterpart, it did instil something extra in the Flemish part, since they weren't only dominated, their culture was also looked down upon. Moreover every side in this example all shares or used to share (when translated to the present) the same religion.