I believe you become Sheogorath in the sense you pretty much rule the SI however I don't think the CoC became exactly like him. It's not the daedric prince that has to change for the realm, it's the realm that has to change for the daedric prince, the realm is completely dedicated to whatever the prince's sphere is. To illustrate, back when Jyggalag was a daedric prince, the Shivering Isles and it's citizens were completely dedicated to order, when he became Sheogorath the Daedric Prince of Madness the Shivering became completely dedicated to madness. So what is to become of the Shivering Isles depends on the CoC's sphere.
I believe it's a little bit of this and a bit of becoming more like Sheogorath. I do believe that Sheogorath's goal with acquainting the Champion of Cyrodiil further with the Shivering Isles, by learning about the two sides, lighting the Flame of Agnon, going through the Ritual of Accession, rebuilding the gatekeeper even, acquiring the Symbols of Office (it was said that the symbols are very important), so on and so forth, was to further integrate the CoC with the realm, to make him/her more like himself (Sheogorath), in a way. "Walk like him to become him" if you will. He wasn't about to give his realm to anyone, especially someone that would compromise what the Shivering Isles was about (at least one could assume). So I believe Sheogorath's goal with all of that was to create a champion who could, for all intents and purposes, take his place and BE Sheogorath.
But also, I don't think the Champion's own self would be entirely compromised, either. Sure, he or she would more than likely be somewhat different after all is said and done (more than likely insane to some degree), but I don't think the CoC's identity would "dissolve" and they'd be completely 100% Sheogorath, as he was before he turned into Jyggalag. A new Sheogorath, if you will. More a myth-echo of Sheogorath than actually Sheogorath, I think.
But maybe I'm totally off base with all that. Just my 2 cents.