Nothing in-game will carry over from one match to another (aside from perhaps stats, which in turn can possibly give you some sorts of aesthetic options. A lot of the fun comes from starting with a fresh slate to work with, choosing this set of abilities or buying those certain items differently. If you were to do matches that have players keep the same character, though you would lose the "clean slate" appeal, you would be able to rush right into the fray with many flashy abilities, which might be neat for some quick matches.
If something like this were to go in, I don't think that they would give you a persistent Demigod, I think that they would give the hosts the options to set a starting level and allow the players to set up a Demigod in the lobby with what skills they would like to start with before the match starts.