About a year and a half ago, I was at a talk by Martin Hyland where he suggested that the Jacobi identity is to the associative law as the anticommutative law is to the commutative law. I think this was in the context of some kind of duality for operads, but I didn't understand at the time.
More recently, I've come to understand that the associative law and the Jacobi identity are essentially the same in the following sense: they both make self-action representations possible. Indeed, the associative law says $$(x \cdot y) \cdot {-} = (x \cdot {-}) \circ (y \cdot {-})$$ and the Jacobi identity says $$[[x, y], {-}] = [[x, {-}], [y, {-}]]$$
Question. Is there a way to make this precise in the language of universal algebra or (enriched) category theory, and are there other examples?
This isn't an answer to your question, but this analogy has always annoyed me because the action of a monoid (say) on itself by left multiplication is always faithful, but the action of a Lie algebra on itself by left bracket is generally not.
A better analogy is to think of the action of a group on itself by conjugation, which is not always faithful but on the other hand preserves group structure. Analogously, the left bracket is a derivation for itself (and this is also equivalent to the Jacobi identity). Indeed, differentiating the conjugation action of a Lie group $G$ exactly gets you the Lie bracket on $\mathfrak{g}$.
This vaguely suggests that you might want to look at the literature on quandles.