I have both devices Rift S and Quest. I bought a game in Oculus Quest, but I cant't play it in RIft S (It looks need to buy again). Software is Beat Saber. In my orders page, I can find it. Ofcause I signin as the same user.
Some games support cross-buy between Rift and Quest, but Beat Saber is not one of them (this is up to the developer to decide). The DLC is cross-buy, however, so if you bought the extra song packs on the Rift store, you will also have them on Quest (once you buy the game on that platform).