The quest
content is almost always canon. The Thieves Guild stories all happened, too, as did Fighters and Mages Guilds... presumably it's the same in Morrowind (although things might get hazy where quest chains conflict).
Whether or not it was the same PC that does everything else is often left vague; I usually have 2 or 3 characters per TES game- 1 for the Main Quest and one or two Guilds (b/c the Main Quest usually makes them BAMF's, so why not?), and then another 2 for the other quests that don't quite fit the character of Persona Prima, either playstyle or morality.
Skyrim reopens a can of worms, though. My PC slaughtered the Dark Brotherhood, and will have nothing to do with either Companions or Thieves. I plan on making 2 additional characters to see those plots through, but what about the DB quest line?
Spoiler Does the Emperor still die?
Even though I've wiped out the group that makes it happen, it would be hard to go forward without having some kind of side-reference to the events, wouldn't it?