In Jyggalag's case, I think that "order" refers to logical deduction and determinism. This is not unreasonable considering that his Great Library contained formulae that allowed him to predict all events before it happened.
According to the Librarian, Dyus:
The great library was the height of logic and deduction. Contained
within its walls were the logical prediction of every action ever
taken by any creature, mortal or Daedric. Every birth. Every death.
The rise of Tiber Septim. The Numidium. Everything. All predicted with
the formulae found within Jyggalag's library.
When the jealous Daedra cursed Jyggalag to live in opposition to all he stood for, they made him the Prince of Madness which is the opposite of logical deduction. In Shivering Isles, Dyus mentions that Sheogorath burned the library because it was the anti-thesis of his new beliefs in personal choice:
When Sheogorath discovered the library he had it burned, insisting
that it was an abomination and that personal choice defied logical
prediction.
Thus, I think that the Lore strongly suggests that Jyggalag is the Daedra of Order in the sense of Logical Deduction and determinism.
Regarding Peryite, there are two main in-game sources of information on the Daedra:
Firstly, we have On Oblivion by Morias Zenon
...Molag Bal elects the employment of other daedra, and Boethiah
inspires the arms of mortal warriors. Peryite's sphere seems to be
pestilence, and Vaernima's torture.
In preparation for the next instalment [sic] in this series, I will be
investigating two matters that have intrigued me since I began my
career as a Daedra researcher. ...
Secondly we have The Book of Daedra
Peryite, whose sphere is the ordering of the lowest orders of Oblivion, known as the Taskmaster.
Neither of them indicate that he's a Daedra of "Order" or that "Order" is part of his sphere. Rather The Book of Daedra makes him sound more like a foreman who manages and "orders" the lesser Daedra around. This is the reason why the primary sources are quick to elaborate that whatever ordering Peryite does is more akin to that of a "Taskmaster".
Notice that in the original text, they add emphasis to the fact that he's a "Taskmaster" by capitalizing the word, but do not do the same for "ordering". This is done to illustrate what his main function is and how he's different from Jyggalag.
Thus, even if we choose to believe that Peryite is a Daedra of Order, it seems clear to me that his sphere of "Order" is distinct from that of Jyggalag's.
Of course, if this is still unacceptable, we could take the cop-out route and simply claim that Peryite took over his Lord of Order position after he became the Prince of Madness! Since Jyggalag was cursed before recorded history, this explanation isn't contradicted by any in-game manuscripts.
I haven't played the older Elder Scrolls games, so I can't answer definitively, but in Skyrim, it seems clear from the Sheogorath daedric quest that they are distinct individuals: they talk to each other and have dissimilar voices.
After further research on the UESP wiki, it turns out that Sheogorath worship has existed for at least a thousand years in Morrowind, but Pelagius III was born in 3E 119, which is 314 years before the Oblivion Crisis and 500 years before the events of the Skyrim/Dragonborn saga. Also, the old Sheogorath was Jyggalag, the Daedric Prince of Order, cursed by the other Daedric Princes. The Hero of Cyrodil takes over the role, becoming the new Sheogorath. So lorewise, Sheogorath shouldn't be Pelagius, and definitely can't be Pelagius after the events of Oblivion.
Best Answer
Well, the answer to your question lies largely with who Shor really is. In other regions, he is known as Lorkhan. Lorkhan was brought about by Padomay, or Sithis, and in turn, brought Mundus (creation) into being by convincing the Aedra to create it. Mankar Camoran maintained that Lorkhan was a Daedric Prince, however this is not confirmed by other lore, and is somewhat odd, considering the Daedra were the Ancient Ones who refused to create Mundus. However, Daedra are traditionally allied with Padomay, like Lorkhan. If Lorkhan really is a Daedra, then it is almost certain that Sovngarde is his plane of Oblivion, comparable to Nocturnal's Evergloam or Hircine's Hunting Grounds. Most of the lore I have seen supports this explanation. As to why Sovngarde is limited to Nords: Lorkhan has long been seen as a supporter of man and enemy of Mer. Since Nords are the original men of Tamriel, so to speak, it is likely that Lorkhan holds a special fondness for them, and their entrance of change into the Mer dominated world. Lorkhan has long been represented as one who introduces change, representing Sithis' realm of chaos.
In conclusion, I do not think a concrete answer has been given in the Elder Scrolls lore as to who Lorkhan is, or what Sovngarde is. The clues seem to point to it as a realm of Oblivion, however.