Because the Shivering Isles is not a direct continuation of the Oblivion Crisis / Martin quest, I don't believe that the Champ of Cyrodiil and the new Sheogorath are necessarily the same person. The Broken Steel DLC for Fallout 3 is a direct continuation of the main quest and there is no question that the Lone Wanderer you played in the original game is the same guy performing those new deeds. In contrast, the content added by Operation Anchorage, The Pitt, etc... can be completely ignored, but following the end of the original Fallout 3 main quest, Broken Steel picks right up and adds another chapter to the continuing story.
Spoiler Canonically, even if you chose to sacrifice yourself at the end of Fallout 3, Broken Steel establishes that your character managed to survive. You don't necessarily have to complete the rest of the main quest in Broken Steel, but by virtue of the DLC's existence your Lone Wanderer is considered alive and well.
Now, the Shivering Isles to me feels like one gigantic side-quest. It's always the main quest that gets the canonical spotlight. Could your character have been the Gray Fox, Listener of the Dark Brotherhood, a vampire, Sheogorath, the Divine Crusader, Arena Grand Champion, Fighter's & Mage's Guildmasters, and Champion of Cyrodiil all in the course if one game? Certainly, but is it canonical that the
same person performed all those amazing feats? Why the hell didn't they make that guy Emperor?
The beauty of an Elder Scrolls game is you can forego the main quest in favor of doing something else, like joining a guild, exploring and doing odd jobs, training up your skills, or endlessly dungeon-diving. Maybe you take the main quest as far as Jauffre. You give him the Amulet of Kings and he tells you to go find Martin and you ... accidentally step too far out onto Dive Rock and plummet to your pathetic death. Eventually some other globetrotting warrior will happen along Weynon Priory and take up Jauffre's quest and
he becomes the Champion of Cyrodiil. Similarly, you could step out of the sewers and head straight for that Strange Door. Personally I only ever finished the main quest in Oblivion once. Every character afterward has completely ignored it, but many of them have spent time in the Shivering Isles.
There's no reason to assume that the Champion of Cyrodiil
isn't the one to become the new Sheogorath, I'm just saying it's not necessarily so. In terms of TES:IV the
game, the Shivering Isles expansion is meant to add more for the consumer who is playing the title Bethesda designed to do. In terms of Elder Scrolls lore it means the Jyggalag/Sheogorath cycle has ended and someone new has taken up the position of Prince of Madness. Like I said above, because the Shivering Isles storyline does not directly affect the main Oblivion storyline I don't think that the CoC is canonically the new Sheogorath.
I think the more interesting question is what will become of the new Sheogorath at the Shivering Isles? Jyggalag says,
"Perhaps you will grow to your new station." which to me foreshadows that the new Sheogorath, CoC or not, is going to show up in a future Elder Scrolls title and be just as loony as his predecessor. In fact, considering that no one outside of the Shivering Isles even knows the old Sheogorath is gone, I think will pretty much be business as usual when it comes to worshipping the Prince of Madness. Will he look or act different? Who knows? Maybe "growing into your station" means you'll become so much like the old Sheogorath that you'll be virtually indistinguishable. Perhaps given enough time you will become one with the Shivering Isles themselves and achieve full Daedric status.