This ties in all of the theories above, and explains everything about the games.
The bunny looks like just a savepoint character, but how does it even posses that power? Well, in The Witch's House, the Cat demon had the power to reverse time to an earlier savepoint. A similar thing is going on here. Mobius makes a pact with the girls, and creates a "Groundhog Day" Loop that lasts until the girl is satisfied, which usually happens in the Golden Ending. This also explains why all the girls are Yanderes; Mobius either corrupts them through the deal or deliberately seeks out those kind of girls to contract with. And it explains why the girls in Yanderella knew about the loops; they both made a pact with Mobius and were aware of the details.
Now as for what the bunny demon was doing in the forest; its hinted that the Red Riding Hood thing was some kind of human trafficking operation, so perhaps thats where the demon originally lived, and its job was to make sure that both the grandma and the woodsman/kidnapper got a steady supply of victims. That is, until the mother decides to rebel, and gives her soul so that one girl may end things. So the bunny protects the blonde girl till the Golden Ending, because Mobius is a demon bunny of thier word.
Now, why do all the protagonists look the same? Perhaps theyre escapees from the trafficking operation. Or perhaps theyre brothers living in the same area, maybe even from the family that was running the trafficking operation. Or perhaps theyre all the same person who gets thier memories erased by Mobius every time a game/loop is completed. Either way, Mobius seems to know them, since they offer their save point services to them.
Now as for why Mobius is doing all this, thats a mystery. Possible motives:
- Theyre simply toying with people For the Evulz and/or for thier amusement.
- Theyre trying to get everyone to kill each other so they can have souls to eat.
- Theyre a particularly twisted Shipper on Deck for the Yanderes.
- Its all part of a bigger plan, and we have yet to see the endgame.
- Some combination of the above.