Can people explain all about this? I want to learn, as I am replaying Oblivion. Are they the same person?

There are some other theories and hypotheses as to what happened, but this seems to be the most commonly held interpretation:

The CoC became Sheogorath. Jyggalag departed from the realm to build a new one, and 200 years later, at the time of TES V, the CoC has, as Jyggalag predicted, grown to his station, now nearly identical to the previous Sheogorath, down to the wardrobe, voice, mannerisms, and accent, although he still occasionally seems to reminisce on his mortal adventures during the Oblivion Crisis.

The Shivering Isles expansion has you work for Sheogorath in his Alice in Wonderland-esque environment.

Well, the American McGee version.

The ending of the game has Sheogorath pass his kingdom to you as he becomes another Daedric Prince.

(Being originally a cursed one).

Basically, it's the Gray Fox plot only you become a god instead of the GF.