Forget for a second during this match that you're the character you've been up until this point.
It really helps.
Forget that you're a Breton Battlemage or a Khajiit Acrobat or an Imperial Agent or a Dunmer Nightblade or a Bosmer Archer or whatever in the world it is that you are. Forget what level you are on, what your skills are, and what enemies you've had troubles defeating in the past.
At this moment in time, you are Lord Sheogorath, a Daedric Prince. If anyone can defeat a Daedric Prince, it's another Daedric Prince. Especially as it's your own realm and you have the home field advantage. I'd imagine Jygg's powers are greatly diminished in a Realm of Madness for he is Order itself.
Keep in mind he's never actually fought against Sheogorath one on one before. He's always easily defeated all the denizens of the Shivering Isles because even in a presumably weakened state he is a Daedric Prince and there are no others on his level to challenge him.
And since this is his own realm, or at least that of his alter ego up until that point, yes, it is the real guy, not some avatar. Sheogorath transformed into him before your very eyes, so this is easily verified.
Nevermind that you could step right back into the world and be killed by a Grummite or a Goblin or something after this epic battle. That's sheer game mechanic. (In my personal opinion the player should actually have become rather more godly once attaining his Godhood, but I understand the limitations of the levelling engine so it's a minor squabble.)
Whole-heartedly agree. And, to expand somewhat on that, the "stronger" the Greymarch it seems(Or how much of Madness it takes over) is how much stronger Jyg's powers are. So, given you've halted the Greymarch, and all those which were, more or less "Focus sites", Jyg's power is weakened. I think it's mentioned somewhere, also, that Daedric Princes are stronger in their own planes of Oblivion than any other place, if I am not mistaken. If I am, it makes a lot of since, even if.