While there is cross-buy between PS4 and PS5 as well as Xbox Series X and Xbox One, if you wanted to transfer your character from PS5 to Nintendo Switch you'd have to buy two separate copies of the game. Plus, you will have to purchase Diablo 2: Resurrected on each individual system.
In order to do this, you will need to have your account linked on all the systems that you wish to carry progression across. No matter what platform you're playing on you are able to transfer your characters and loot from one platform to another. These consoles are the PS4, PS5, Xbox One, Xbox Series X, Xbox Series S, and Nintendo Switch. It was confirmed prior to release that players can enjoy cross-platform progression between PC and consoles.