Your character from TES4 was still mortal, and would presumably have died long before TES5. I'm not sure why Sheogorath still looks the same as he did in the last game though...
Except, he totally doesn't?
Oblivion Sheogorath
http://images.uesp.net/e/e1/SI-Sheogorath.jpg
Skyrim Sheogorath
http://images.uesp.net/5/5e/SR-npc_Sheogorath.jpg
And these two Sheogorath's look nothing like the one from Daggerfall/Morrowind.
Remember, before Skyrim it could have been a possibility that the CoC just went insane or the like and never actually became Sheogorath, but in Skyrim Sheogorath specifically references the fact he/she was once the CoC. The CoC simple has mantled Sheogorath so much that he/she has become Sheogorath, even in how he/she looks like.
That's how I always thought of it. After roughly 200 years, my Redguard started to mantle Sheogorath so well that he eventually began to look like Sheogorath. Still different, but similar.
God of madness is a family title sheogorath hands down to himself every few hundred years.
He says that himself during his daedric quest, right?