Haven't played the Shivering Isles questline to the end but:

"and even visited http://www.uesp.net/wiki/Lore:Sheogorath's realm of insanity to battle the Daedric prince and end an eons-long cycle of order and chaos, becoming the new Sheogorath, Prince of Madness." -UESP

Now I'm officially confused. What happens here? Does the hero in Oblivion actually kill Sheogorath and replace him, or merge with his oversoul or something? And if so, if the Sheogorath we see in Skyrim Sheogorath, or the hero from oblivion?

The champion defeats Sheogorath (http://www.uesp.net/wiki/Lore:Jyggalag), then becomes Sheogorath, thus freeing Jyggalag from being Sheogorath, and return to only being Jyggalag, the prince of order.
So, yes, there is a possibility that the Sheogorath you see in Skyrim is actually the Champion of Cyrodiil.

Shivering Isles main quest line consisted of Sheogorath's attempts to groom the player character into a worthy successor to the Throne of Madness, banking that, with a "new Sheogorath" on the throne during the Greymarch (where Sheogorath returns to his original form, Jyggalag, and destroys the Isles in the name of order, leaving Sheogorath to rebuild), Jyggalag could be defeated.

Whether or not the Champion is Sheogorath in Skyrim is, like anything on the lore forums, debated, but it seems that this is what Bethesda is going with.

That's awesome.