I'm wondering if this notation "makes sense" and one can substitute $Z$ for other fields, rings and get other structures. Is this notation common, can one point me to resources using it?
For example this notation appears in this wikipedia section 3 times, in this paragraph for example (latex not copied literally, to see it directly click on the link is the second paragraph).
Such a ring homomorphism R → EndZ(M) is called a representation of R over the abelian group M; an alternative and equivalent way of defining left R-modules is to say that a left R-module is an abelian group M together with a representation of R over it.
Also in this textbook in spanish, page 99, just under definition 3.1.1.
In general, if $R$ is a ring, and $M$ is an $R$-module, then $\mathrm{End}_R(M)$ denotes the ring of $R$-endomorphisms of $M$; in other words, it is the collection of $R$-module homomorphisms $M\to M$.
Abelian groups are the same thing as $\mathbb{Z}$-modules, and if $G$ is an abelian group, then the group endomorphisms of $G$ are the same thing as its $\mathbb{Z}$-module endomorphisms, $\mathrm{End}_{\mathbb{Z}}(G)$. For any ring $R$, an $R$-module $M$ is also a $\mathbb{Z}$-module (because there is a (unique) ring homomorphism $\mathbb{Z}\to R$) and therefore, considering an $R$-module $M$ as a $\mathbb{Z}$-module, i.e. an abelian group, $\mathrm{End}_{\mathbb{Z}}(M)$ are its group endomorphisms.