I will expand on my comment. It seems to me that such colorings are tightly related to 2-factorizations of graphs. Indeed every color class represents a subgraph which is a disjoint union of directed paths and cycles. Let's denote by $C(x)$ the set of edges colored $x$. We can see that the underlying graphs of $C(x)$ and $C(-x)$ coincide, but they have opposite orientation.

It's possible to show that a graph with $n$ vertices and maximum degree $\Delta$ can be written as a union of at most $\lfloor \frac{\Delta+1}{2}\rfloor$ subgraphs of maximum degree 2. Each of these subgraphs is a union of paths and cycles, so we can assign two colors to the corresponding directed edges. This shows that there is always a coloring satisfying the properties in the question coming from $\mathbb Z/(\Delta+1)\mathbb Z$. This would be the most straightforward analog of Vizing's theorem, and $\Delta+1$ cannot be improved.

It seems to me that such colorings are closely related to 2-factorizations of graphs. Indeed every color class represents a subgraph which is a disjoint union of directed paths and cycles. Let's denote by $C(x)$ the set of edges colored $x\in G$. We can see that the underlying graphs of $C(x)$ and $C(-x)$ coincide, but they have opposite orientation. When $2x\neq 0$ then the underlying subgraph must have maximum degree 2. When $2x=0$ then $C(x)$ is a matching.

It's possible to show that a graph with $n$ vertices and maximum degree $\Delta$ can be written as a union of at most $\lfloor \frac{\Delta+1}{2}\rfloor$ subgraphs of maximum degree 2. Each of these subgraphs is a union of paths and cycles, so we can assign two colors to the corresponding directed edges. This shows that there is always a coloring satisfying the properties in the question coming from $\mathbb Z/(\Delta+1)\mathbb Z$. This would be the most straightforward analog of Vizing's theorem. Notice that $\Delta+1$ cannot be improved.