Every Abelian group is canonically a $\mathbb{Z}$-module, where $\mathbb{Z}$ is the initial monoid in the monoidal category of Abelian groups. And every Abelian monoid is canonically an $\mathbb{N}$-(semi-)module, where $\mathbb{N}$ is the initial monoid in the monoidal category of Abelian monoids.
Are these two examples special cases of a more general principle? Or is the similarity between them just a coincidence?
If $C$ is a closed monoidal category and $R$ is a monoid object in $C$, then a left $R$-module is the same as an object $M \in C$ together with a homomorphism of monoids $R \to \underline{\mathrm{End}}(M)$. If $R=\mathbf{1}_C$ is the initial monoid, it follows that every object of $C$ has a unique $R$-module structure. If $C$ is not closed, it can be also checked directly that every object has a unique left $\mathbf{1}_C$-module structure.
For $C=(\mathsf{CMon},\otimes,\mathbb{N},\dotsc)$ we see that every commutative monoid has a unique $\mathbb{N}$-module structure, and for $C=(\mathsf{Ab},\otimes,\mathbb{Z},\dotsc)$ we see that every abelian group has a unique $\mathbb{Z}$-module structure.