Well, to bei honest i never heard of this version of chess but i like the concept.
Teleport chess

Is it only the king or it could be any piece if it meets the material value, distance, or any other requirement?

its only the king and to be clear it has to be touching one of your pieces and if you switch before castling you lose castling rights. also you cant switch out of (or into obviously) check other wise end games take forever. its pretty fun setting up traps with the ability we got the idea watching naruto
I had this idea to play a game of chess with the home rule that the king can switch places with any piece it touches as well as regular castling, me and my wife have been playing it and having a lot of fun. Is this called anything? Have any of you played a home rule game like this before?