Noteworthy examples of finite categories

One example is what is called a fusion system.

A fusion system is a category where the objects are the subgroups of some fixed $p$-group $S$ and where the morphisms is a subset of the set of injective homomorphisms between the subgroups which contains all those induced by conjugation by some element from $S$.
Further, it is required that any morphism $\varphi$ from $P$ to $Q$ factors through the inclusion of $\varphi(P)$ into $Q$ and that the inverse homomorphism $\varphi^{-1}: \varphi(P)\to P$ is also in the category.

These are meant as a generalization of the fusion structure on the set of subgroups of a $p$-Sylow subgroup of a finite group, and they have been studied extensively over the past 10 years or so, especially by people in various areas if algebraic topology.

I like the Mathieu Groupoid $M_{13}$.

Also, for any group $G$ acting on a set $X$ there is the action groupoid $X/\!/G$ with objects the elements of $X$ and with morphisms $x_1 \to x_2$ given by the elements of $G$ such that $g \cdot x_1 =x_2$. (Of course, if $G$ and $X$ are finite, so is $X/\!/G$.)

(Maybe these don't quite count, as any groupoid is equivalent to a disjoint union of groups and you already mentioned monoids...)