The leash can obviously moves through the doors since it can move upwards, so if enough pressure (not that much) were applied to the leash it would easily pull out, granted the owner stopped holding it. Elevator doors aren't that hard to pull open.
are you saying that if the owner let go of the leash then the handle would have slipped out of the elevator door and fell down the shaft dropping the dog?
the handle is thicker than the leash, that's a given
so the leash part pops out of the probably rubber handle and falls down the shaft