Kurizted

It would be interesting to play classic games and, on top of that, have a trophy list. It's something we've already seen with ported games, like Final Fantasy VII, Bully, Siren, etc. so there's hope that they can include trophies. But if they don't, it's not a problem either.

Just levels, cosmetics and the Pass will cross-save, but not trophies progression. So yes, you're correct.