It would make no sense if you can change the race/class of your companions? I mean how do they even design reactivity/story around that possibility? It would simply be impossible. So much depth would be lost in that case.
I mean, the same can be said if your own character can go from being a human lvl 1-5 and then suddenly be a tiefling. There is absolutely no way the game is designed to react to those changes.
