There is a much older custom that seems so similar it is hard to imagine that it is not related. When passing an item to someone - say a cup to a house guest - one is traditionally never supposed to hand the item directly to the other person - i.e. never directly from hand to hand. Typically this meant placing the item on a tray (お盆) and offering the item on the tray to the person (or at least placing the item in front of the person for them to pick up). You can see examples of this today in, for example, formal Shinto ceremonies a ceremonial square tray is often used (折敷) or frankly at any fancy kaiseki restaurant.
I'm not sure about the origin of the custom, but perhaps the action of taking something out of someone's hand feels too close to taking it forcefully from them - i.e. for the same reason it is considered impolite for two people to hold the same piece of food by chopstick (for example, if passing food between people from chopstick to chopstick).