Did the mortal whom stopped the Greymarch literally transform into the new Mad God overtime, or was it more like being possessed/ serving as a vessel?

No possession which I can see.

The whole of the various Shivering Isles Quests seems to be all about powering up a new Madgod by Sheogorath, so he can be free to be his original self.

It also opens up the concept that Daedra can be made.

Skyrim displays Sheogorath retaining full memory of the Oblivion crisis.

I think Beth did a nice piece of work on that though, considering the new Sheogorath would be markedly different from the old in many ways, even including how he/she would appear physically.