Realizing a monoid as $\mathrm{End}(G)$ for some graph $G$

Trying to forestall another question along these lines let me add that every locally presentable category fully embeds into the category of graphs - see Adámek, Rosický Locally presentable and accessible categories 1994.

In fact, depending on set theories in which we work, every concrete category fully embeds into graphs - see Pultr, Trnková, Combinatorial, algebraic and topological representations of groups, semigroups and categories 1980.

For example: the category of metrizable spaces embeds into Graphs and in some set theories the category of Hausdorff topological spaces also embeds into Graphs.

Monoid is a category with a single object, hence a special case of the first result.

Hedrlín and Pultr proved that every monoid was the endomorphism monoid of a graph. See their paper "Symmetric relations (undirected graphs) with given semigroup" Monatsh. Math 69 (1965), eudml, DOI: 10.1007/BF01297617.