I think you could phrase this question purely algebraically in terms of $\mathbb{F}_q[x]$ acting on $\mathbb{F}_q$ by evaluation, but since $\mathbb{F}_q[x]$ doesn't form a group under composition I wasn't sure how to do that properly. Instead I formulated it in terms of graph theory.
Given a polynomial $f \in \mathbb{F}_q[x]$, we can define a labelled directed graph $G_f$ with vertices $\operatorname{V}(G_f) = \mathbb{F}_q$ and with edges $\operatorname{E}(G_f)=\{a \to f(a) \mid a\in\mathbb{F}_q\}$. Such a graph must be a functional graph. For which $g \in \mathbb{F}_q[x]$ are the graphs $G_f$ and $G_g$ isomorphic? What about isomorphic up to relabelling of vertices? In general, how does $\mathbb{F}_q[x]$ become partitioned under this relation based on acting "identically" on the underlying field elements?
I would hope that there is something nice and clean that can be said to answer this question, but otherwise I would be quite happy with some references to similar research, or even just some vocabulary to Google to learn more about problems like this.