To what extent is Momonga living in the current world in which this anime is set? Did the human body of Momonga transform itself into his character in the game? If that's the case, his soul should be in this character.

Or is it something similar to SAO in which his original body is still in the "human world", but his soul (or consciousness) in the "game world"?

Or could it be that Momonga is with his soul and human body in his own world and sits in front of his computer playing this realistic game?
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.