What's the meaning of 'off' in this context?

Question

"But we don't feet like leaving, do we, boys? We've eaten all our food and you still seem to have some."
Goyle reached toward the Chocolate Frogs next to Ron - Ron leapt forward, but before he'd so much as touched Goyle, Goyle let out a horrible yell.
Scabbers the rat was hanging off his finger, sharp little teeth sunk deep into Goyle's knuckle - Crabbe and Malfoy backed away as Goyle swung Scabbers round and round, howling, and when Scabbets finally flew off and hit the window, all three of them disappeared at once.
(Harry Potter and the Sorcerer's Stone)

Scabbers the rat still attached to Goyles’s hand. And yet why the preposition off, that means coming off the finger, is used?

Answer 1

When a person or item is "hanging off (something)" in the sense used, it is attached to that (something) at or near its upper end, while its lower end is loose and can easily flap or flail about. In the example sentence, Scabbers is attached to the finger via his teeth, while his body and legs are unsupported. "Hanging off" generally has the connotation that the hanging item is not firmly connected and may be subject to an undesired, potentially damaging fall.

"Hanging on (something)" is a much more generic description, with the implication that the item being hung is firmly or appropriately attached to the (something). Generally, if a fall is possible, it is either very unlikely under normal circumstances or will not be damaging. You can hang a picture on the wall, but you can't hang a picture off the wall because the bottom of the picture won't be flapping free (it'll be against the wall, not moving). You can hang your coat on the coathook, but you can't hang your coat off the coathook (because the coathook is designed to have coats hanging on it). Also, in contrast to "hanging from" (see below), "hanging on" does not imply that the hanging object is mostly below the supporting thing; more likely it will be beside it or around it.

"Hanging from (something)" is similar to "hanging off" in that it implies that the lower end of the hanging object is loose, and more specifically that a significant portion of the hanging object is below the lower edge of the supporting something; but it doesn't have the same sense of an impending fall. I would say, for example, that "a banner hanging from the top edge of a building" implies that it was placed there on purpose and properly secured, while "a banner hanging off the top edge of a building" implies that it was placed on the roof, and accidentally got blown partway over the edge.

"Hanging by (something)" is used to describe the element connecting you to your anchor point: Scabbers is hanging OFF a finger, BY his teeth; a chandelier hangs FROM the ceiling, BY a chain. This is often used in the idiomatic phrase "hanging by a thread" to describe something that is in a very precarious situation where a disastrous fall is likely at any moment.

These are good general guidelines, I think; but @StoneyB does give some examples that contradict them, which is pretty typical of English.

Answer 2

It's totally futile trying to predict what an English preposition "means" in any given context.

Hang (and depend, and suspend, and dangle) takes on, from and off in practically the same sense.

His coat was still where he left it, hanging on its hook.
A lamp hung from the ceiling on a chain.
He hung a banner off the side of the bridge.

It's also futile trying to predict which preposition is appropriate in any given context. A coat almost never hangs off a coathook; a lamp hangs on the wall but from the ceiling. You just have to take the usage as it comes.