Here is a simple observation: The condition is equivalent to $$\forall a,b \in M. \, [a \cdot [b]]=[a \cdot  b]=[[a] \cdot  b].$$
Assume that $M$ is a preordered monoid. Then it is natural to assume $a \leq [a]$, and $a \mapsto [a]$ behaves like a "closure operator". The fixed points are the closed elements. This setting can be generalized and has been studied before:

Assume that $M$ is a monoidal category with underlying category $C$ and $R : C \to C$ is a functor equipped with a natural transformation $\eta : \mathrm{id}_C \to R$. Then one may demand that the induced morphisms
$$R(a \otimes R(b)) \leftarrow R(a \otimes b)  \to R(R(a) \otimes b)$$
are isomorphisms. This situation appears in Day's [reflection theorem][1] for closed monoidal categories; here the reflection is called **normal**. This is used to endow reflective subcategories of $C$ with a monoidal structure. It is also useful for the construction of monoidal localizations, see Day's [Note on monoidal localization][2]. I would also consult the papers which cite these.


  [1]: http://www.sciencedirect.com/science/article/pii/0022404972900217
  [2]: https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0004972700045433