Something like this?
The Prince of Madness prior to SI was named Arden-Sul.
The PC becomes a Prince of Madness named Sheogorath....and BAM...Sheogorath has been the Prince of Madness from the beginning.
Just like the tatters of the "true tale" of the Tribunal still flitted around in Morrowind, so do the lost remnants of the old Prince peek out through the notion of the dead Arden-Sul. Since a Prince of Madness is both Mania and Dementia, there are two tales about this great guy.
Now, we all know that "Sheogorath" has been the Prince through all the Elder Scrolls games,...but do we (as the PC) simply remember it that way now?
Reminds me of when the Gray Fox says "I just told you my real name three times, but you don't remember it."
The in-game books Zealotry of Sheogorath and Heretical Thoughts allude to Sheogorath having a mortal aspect named Arden-Sul and being a flesh and blood mortal respectively. Look at it this way, and, mind you, this is just an off-the-wall thought that I had and wanted to share:
What occurred in SI was that the Prince of Madness, Arden-Sul chose you to become his champion and [so on and so forth]. Arden-Sul was the Madness persona to Jyggalag's Order. The PC comes along and completes the MQ of SI, thereby doing the mantelling and all that. What happens at the moment of mantelling is that, all of a sudden, no one remembers that "Arden-Sul" was the Daedric Prince...it is now you, Sheogorath, and always has been in this new timeline. The name "Arden-Sul" is not completely erased from memory, though. In fact, some people remember the old Prince as a Duke of Mania. Some remember him as a Duke of Dementia. Some, such as the Zealots, remember him as the mortal (because he's dead now) aspect of "Sheogorath" (the Prince from the beginning in their eyes). Others, such as the Heretics, remember you, the PC, slightly from the pre-mantelling days. Something subconscious perhaps that makes them think that you are NOT a true god, that you are a mortal driven mad from consorting with the Daedra, which, by the way, is exactly what happened to the PC.
Don't forget...."Good Gods come and go, but all Lords eventually fall. A God can wake up mortal." The Ravings of Fenroy
The four notions presented by the in-game books are:
Mortal
Mania Duke
Dementia Duke
God
They represent all the possible stages in SI. Reminiscent of the Daggerfall Dragon Breaks.
So complete is the Break that even you, the PC, remember the Prince of Madness as "Sheogorath", although you do remember being mortal.
This theory doesn't touch on the Sheo=Jygg problem though, just what happens/happened to the Prince of Madness.
One final thought. In the text of Ravings of Fenroy, the verse goes like this:
Just You wait and see
Good Gods come and go, but
All Lords eventually fall
A God can wake up mortal.
Notice that all the capital letters spell "JYGGALAG"? I don't believe that that is a coincidence.
___The Word Merchant of Julianos