The category of Lie algebras is equivalent to a certain category of cocommutative Hopf algebras, with the equivalence given by sending a Lie algebra $\mathfrak{g}$ to its universal enveloping algebra $U(\mathfrak{g})$. These cocommutative Hopf algebras can in turn be thought of as group objects in a certain category of cocommutative coalgebras, and hence can potentially pop up as automorphism objects in any category enriched over cocommutative coalgebras.

You might object that you don't know any interesting examples of such categories, but in fact you do: the category of *commutative algebras* admits such an enrichment (see the nLab), and this is one abstract way to see why Lie algebras can act on commutative algebras (by derivations).

Speaking more philosophically, you should expect to be able to extract Lie algebras from any situation where you can cook up a sensible notion of infinitesimal automorphism or more generally an infinitesimal element of some group. Enriching over cocommutative coalgebras gives you one fairly general way to do this; if $X$ is an object in your category and $\text{End}(X)$ is the cocommutative bialgebra of endomorphisms of $X$, then the primitive elements of $\text{End}(X)$ (the ones satisfying $\Delta X = 1 \otimes X + X \otimes 1$, where $1 = \text{id}_X \in \text{End}(X)$) should be regarded as the infinitesimal endomorphisms of $X$, and indeed these naturally form a Lie algebra under the commutator bracket.

property. – Hiro Lee Tanaka Jul 25 at 12:02linear structure. (Additive is a good place to start; better if you have something k-linear, or something stable.) In this setting, (coLie)Lie is Koszul dual to (commutative)cocommutative, and Koszul duality for augmented objects gives the interpretation of Lie objects as infinitesimal objects studied "at a point." I don't think the Koszul dual to cocommutative algebras in, say, spaces or sets (with whatever monoidal structure) has such a rich interpretation, though I'd be happy to hear about one! – Hiro Lee Tanaka Jul 25 at 12:11