So when Card X plays Band of Misfits twice, and Band of Misfits leaves play between the two occasions, then Band of Misfits can be two different things if X is Prince but it has to be the same thing twice if X is Throne Room. Right?
TR does not put a card into play twice. It plays it twice, but puts it into play once (the first time). Prince puts the card into play several times, because it explicitly makes the card leave and enter play each time. I guess TR cannot put the card into play again because 1. it would not make sense if the card is in play, where TR left it. 2. it could not put it into play from anywhere else, because TR would have lost track of it.
Thus, what makes Prince able to change BoM is the fact that the card is put into play several times, not the fact that it leaves play in between. I guess while you are playing BoM the second time as Feast, the BoM in the trash is actually a BoM and costs $5.