It will also improve your relations over time with any clan that has non party leading members staying inside the chosen fief. For instance, if you leave your companion inside your faction's capital, then you'll usually start gaining relations with the faction leader, since some of his relatives will be residing in the town's keep.
Companions also have a chance to win tounaments and gain skill points if the player doesn't join and the tournament resolves automatically. If you don't want to do a tournament yourself, you could pop a companion or two in there for a few days to see if they'll win.