If a monoid is a category with a single object, is there a "monoid-theoretical" concept that the categorical product translates to? As an analogue, in a poset the product translates to the notion of meet (and the coproduct to its dual, the join).
Just by unwinding the definition one can see that this boils down to the existence of two monoid elements $p_1$ and $p_2$ such that for every $x$ and $y$ in the monoid there's a unique element $\langle x,y \rangle$ that factors as $x = p_1\langle x,y \rangle$ and $y = p_2\langle x,y \rangle$; the question is whether the concept is a known and/or useful one.
Incidentally, if the monoid in question is a group then the existence of a product makes the whole thing collapse into the trivial group; this might or not hint that the concept I'm looking for is "too boring" to have a name. In fact it suffices for one of the projections to be right cancellable, or for the pair $\langle x,x\rangle$ to be left cancellable for some $x$, to make the whole thing collapse.
I know exactly one interesting thing to say about this, which is the following: as Berci says in the comments, this is equivalent to asking for objects $A$ in categories which are isomorphic to their cartesian squares $A^2$. There is a somewhat more general thing you could ask for, which is to ask for objects $M$ in monoidal categories which are isomorphic to their monoidal squares $M^{\otimes 2}$.
The fun theorem is that the free such thing (that is, the free monoidal category on an object equipped with an isomorphism to its monoidal square) has one nontrivial object whose endomorphisms are the Thompson group F. This is due to Fiore and Leinster.
Hence the Thompson group naturally acts by automorphisms on any object equipped with an isomorphism to its monoidal square (in any monoidal category whatsoever), and in particular there's a natural map from the Thompson group to any monoid satisfying your condition.