Generalisation of the statement that a monoid recognizes language iff syntactic monoid divides monoid Let $A$ be a finite alphabet. For a given language $L \subseteq A^{\ast}$ the syntactic monoid $M(L)$ is a well-known notion in formal language theory. Furthermore, a monoid $M$ recognizes a language $L$ iff there exists a morphism $\varphi : A^{\ast} \to M$ such that $L = \varphi^{-1}(\varphi(L)))$.

Then we have the nice result:

A monoid $M$ recognizes $L \subseteq A^{\ast}$ if $M(L)$ is a homomorphic image of a submonoid of $M$ (writen as $M(L) \prec M$).

The above is usually states in the context of regular languages, and then the above monoids are all finite.

Now suppose we substitute $A^{\ast}$ with an arbitrary monoid $N$, and we say that a subset $L \subseteq N$ is recognized by $M$ if there exists a morphism $\varphi : N \to M$ such that $L = \varphi^{-1}(\varphi(L))$. Then we still have that if $M$ recognizes $L$, then $M(L) \prec M$ (see S. Eilenberg,  Automata, Machines and Languages, Volume B), but does the converse hold?

In the proof for $A^{\ast}$ the converse is proven by exploiting the property that if $N = \varphi(M)$ for some morphism $\varphi : M \to N$ and $\psi : A^{\ast} \to N$ is also a morphism, then we can find $\rho : A^{\ast} \to M$ such that $\varphi(\rho(u)) = \psi(u)$ holds, simply by choosing some $\rho(x) \in \varphi^{-1}(\psi(x))$ for each $x \in A$ and extending this to a morphism from $A^{\ast}$ to $M$. But this does not work for arbitrary monoids $N$ so I expect the above converse to be false then. And if it is false, for what kind of monoid beside $A^{\ast}$ is it still true, and did those monoids have received any attention in the research literature?
Yes, these monoids have received attention in the research literature and actually lead to difficult questions.

Definition. A monoid $N$ is called projective if the following property holds: if $f:N \to R$ is a monoid morphism and $h:T \to R$ is a surjective morphism, then there exists a morphism $g:N \to T$ such that $f = h \circ g$.

You can find a long discussion on projective monoids in [1], right after Definition 4.1.33. It is shown in particular that every projective finite semigroup is a band (a semigroup in which every element is idempotent). But the converse is not true and it is actually an open problem to decide whether a finite semigroup is projective.

[1] J. Rhodes and B. Steinberg, The $q$-theory of finite semigroups. Springer Monographs in Mathematics. Springer, New York, 2009. xxii+666 pp. ISBN: 978-0-387-09780-0