"Sheogorath" is but the ruler of the Realm of Madness, bound to the Tree of Madness. Jyggalag was a Daedra Prince, thus as Sheogorath he remained a Daedra Prince. The CoC was a mortal, thus as Sheogorath he/she remained a mortal.
Sheogorath, "mad
god", lord of the neverthere, the sithis shaped hole of the world. He is not simply a ruler but a god i.e. chaos/madness given a personification which is Sheogorath. Sheogorath is a seperate entity from Jygallag.
When you try to split or break a god many things can happen. In this case Jyggalag was cursed and the personification of madness was formed
from the personification of order. So Sheogorath was formed from Jyggalag and is himself a daedric entity. The tree of madness is merely a symbol that Sheogorath has enabled to function in the material realm, he coould easily have created a cow of madness to achieve the same function, it is no more special than the staff of madness, the throne of madness or any of the other symbols that allow the embodiment of the concept of madness to function. The tree of madness is formed from the idea/concept that is Sheogorath. The fundamental concept (sheogorath/madness sphere) to limited form (tree).
The CoC became Sheogorath the personification of madness. A god is a personification of an idea, and the CoC became the madgod i.e. an immortal embodiment of the idea of madness/chaos, not merely a mortal ruler.