I think having a true ‘class system’ relies on having an aristocracy type level at the top. There is the odd lord or lady of the manor type installed in a stately home style set up throughout ROI / the North, but not enough to count.
So I don’t think a class system exists in either part of the island.
What does exist is a thousand societal quirks and of course there is aspiration, discrimination and judgment along myriad factors and markers of your past and current wealth, occupation, education, social circle. These vary wildly not just between the two sides of the border but from one province or even county to the next. Even within counties, for example North Down and South Down are different beasts. And in the cities, particularly Belfast - any affluent area will sit pretty much side by side with a socio-economically challenged community (East Belfast and Malone have been juxtaposed in this thread but that’s a bit clumsy - Malone / Sandy Row (both in South Belfast) and Ballyhackamore / Connswater (both in East) illustrate the point more neatly).
The role of religion is probably totally phased out in traditional church communities (Catholic and older Protestant churches) and of course also in the growing non religious / non Christian parts of society but it is a whole other matter with the evangelicals.
What I do think is hugely different from England is that humility is absolutely key. People don’t begrudge success but absolutely take a dim view of ‘notions’ (nebulous concept but for these purposes it’s the idea of getting above your station). This applies across all counties I think and is another reason why basically we don’t have a class system (but again emphasising this doesn’t mean people aren’t judging each other!).
I went to a very old, very esteemed university in England with a massive public school contingent, and it was an eye opener in terms of the English upper class. I had no idea. They do inhabit a totally different world. I don’t think there’s a divided off segment of society like that in any part of Ireland, really.