Well, the only wooden, sticky, musical things that aren't instruments that I can think of are a baton (disqualified because they come in animal types and aren't obviously abstract) and the pole that some C18 & 19 conductors would use to beat time.

Oh and I'm quite glad my previous guess was wrong, now I start to think of the 'edible' case.

[INJ] In that case, as we might disagree on the definition and you might have guessed it otherwise, I think it only fair to hand over the BATON to you, hoping to be enlightened about the animal case! The cases of which I was thinking are the wooden (or sometimes, apparently, fibreglass) conductor's baton, the military baton, the metal relay baton, the well-known small loaf, carrot batons and police batons, plus of course the abstract Baton Sinister from heraldry. Challenging subject, but I enjoyed attempting something a bit more hard to get hold of, as it were. Apols to anyone else who had foregone that guess on non-animal grounds, too.

---

They make them in bone and ivory
Anyway, moving on - Let's have a nice simple ANIMAL

They don't come much simpler than that.