That must be part of it, I went into a saved game a leveled a sovereign with 0 max mana, and that did indeed raise his mana cap to the amount of essence I gave him. That doesn't seem like the full story, though, since when my various sovereigns did have mana greater than 0, it was usually not the same amount as their essence. In my example above, when my sovereign had 11/11 mana, he only had 10 essence.
Edit: Upon further investigation, I see that max mana must be essence multiplied by something from one of the sovereign's stats.
Where would I find more information about things like this? I looked through the wiki, but it seems to not have been updated in a few builds.