ToothFairy use case: Sharing a (first-generation) Magic Trackpad between multiple Macs

Could it be that this is the sole reason? Does the Magic Trackpad 2 do things differently?

It would be cool if other owners of an Apple Magic Trackpad of either version could chime in. I only have the original A1339, which does work fine, and nobody to borrow a Trackpad 2 from.