There is a special term for elements of a ring that are multiples of each other: "associates".
In a wider context associates are elements of a semigroup that generate the same ideal.
The equivalence classes of associates appear in multiple places in the theory of rings:
irreducibles, gcds, lcms, etc.
However, the role of associates declines when moving from rings to fields (or from semigroups to groups).
Instead, the role of elements that are powers of each other becomes more visible.
The equivalence classes of such elements generate the same cyclic subgroups.
It looks like they appear in Galois theory as "interchangeable" basis elements in automorphisms of field extensions
(I only started learning the theory).
I am wondering if there is a special term for elements of a ring (or a semigroup) that are powers of each other.
Or, maybe, there is a wider term that generalizes the property?
It's an interesting question, but I think this relation is not much like associates. (The relation I think we are talking about is $a\sim b$ if $a^n=b$ and $a=b^m$ for some $m,n\in\mathbb Z^+$. That's what "powers of each other" sounds like, to me.)
The main weirdness, I think, is that an element is usually not related to its powers. (Contrast this to associates, where unit multiples are always mutually related.)
If $a\neq b$ and $a\sim b$, it would imply $a^k =a$ for some $k > 1$. I've seen "periodic ring" used to mean a ring in which for every element $x$, there are two positive numbers $n,m$, $m>1$ such that $x^n=x^m$. A special case is when $n$ is always $1$. We could say that an element is periodic if $a=a^n$ for some $n>1$. Two elements which are powers of each other would have to be of this type. I found a reference mentioning "periodic elements of a ring" here, but I wasn't able to view it. The references make it look like it is using the definition that I mean.
It seems like this relation can be washed out by a lot of mundane conditions. For example, if you have an $\mathbb N$-graded ring, elements of positive grade are never going to be related to anything but themselves, since their powers have higher grades. The prototype example would be $x$ in $F[x]$.
Even in a very simple periodic ring, it looks like the "powers not related to their base" problem I mentioned seems to be an issue. Consider a ring in which $x^3=x$ for all elements. Then $x^2$ is clearly a power of $x$, but $x$ is not a power of $x^2$ because $x^2$ is idempotent.