You're right, absolutely no way this makes any sense. In so many different ways. For a start, as said above, you can't record backing vocals to a song - "Shining Star" - that wasn't even written at the time. No technology in the world can make that a reality. Yes, of course, if it was just "oohs" and "aahs" but not if one is singing actual words. (Plus the fact that the backing vocals on that record are clearly performed by men. As indeed are all the sixties' Manhattans records I know.)
Elsie Baker sounds a smart, likable and very switched on woman based on an interview I listened to but she also seems to be claiming in that interview that either- it isn't quite clear exactly what the claim is - The Debonaires recorded "Baby Love" before the Supremes or sang the backing vocals on it. That actually makes more logical sense than the Manhattans' story but I still suspect would be news to most people.