To what extent is Momonga living in the current world in which this anime is set?
He's definitely transported into the game, unlike in SAO, all the game features are gone, if you'll recall in the first couple episodes, he has to adjust to his powers and how to use them.  In SAO even though they were "full dive" they still played it like a game, while in overlord it was more, this is real life, not just a really realistic video game.