You are indeed automatically turning into The Lost, and this is permanent for the next victory laps of the same run. This person on Reddit also talked about it and some other details about going down with the victory laps:
At the start of your 3rd lap (so your 4th time on Basement) you are transformed into the Lost. As far as I know, this is permanent. Lost has the same recent items as your previous lap, you don't automatically get Holy Mantle, one hit you're done.
But it doesn't seem like you are unlocking The Lost as a playable character by becoming him with this lap. It seems than the only way to unlock The Lost in Afterbirth is to hold a Missing Poster and to die in a Sacrifice Room.