Periodic functions from $\mathbb{R}$ are frequently studied. Also from $\mathbb{C}$ and $\mathbb{Z}$. It would seem possible to define periodicity for functions from an arbitrary group $G$ to any set $S$ but I cannot easily find much discussion of periodicity in these more general contexts.
If the group is Abelian with the operation $+$ then the definition could be identical:
$$\exists t \in G \ \ \forall x \in G \ \ f(x + t) = f(x)$$
Of course, it will not necessarily be possible to choose a canonical period hence we may need to talk of "a period" rather than "the period".
For a non-Abelian group, we may need to distinguish left and right perodicity.
$$\exists t \in G \ \ \forall x \in G \ \ f(t x) = f(x)$$
$$\exists t \in G \ \ \forall x \in G \ \ f(x t) = f(x)$$
Have I just not searched well enough?
Is this property usually given a different name in these more general contexts?
No one finds it interesting?