Suppose we have a functional question, such as the one here. Often a key step in these problems is to try to equate or chain results together (see my hint). From looking at these questions, it seems that there is often only a finite number of ways that we can attempt to join them together. Can this fact be examined more formally?
Note: I am not asking you to turn my answer from that question into something algebraic. It is that way purposely as I was only trying to give a hint, not a complete solution. What I am asking, is whether there are any results that show that the functions can only be "combined" or "equated" in a certain number of ways on some reasonably general class of problems.
One example is that there's an easy way to get uniqueness of functional equations in certain cases. Consider the "cohomological" equation $f(Tx) - f(x) = g(x)$ (which is important in dynamical systems, e.g. it comes up when you are trying to find an invariant $n$-form). The goal, of course, is to prove that there exists $f$ given $g$. If $T: X \to X$ is topologically transitive, then any two solutions differ by a constant. This is because if $x_0$ has a dense orbit, then the value of $f(x_0)$ determines $f(Tx_0), f(T^2x_0)$, and so on.
More generally, there is the same uniqueness for the "twisted" cohomological equation $f(Tx) - \alpha f(x) = g(x)$, where $\alpha$ is a nonzero real number.
(This is not, I think, relevant to the question, but I'll mention that one can get existence of a Hölder solution in the untwisted case for transitive Anosov diffeomorphisms of compact manifolds if the sum of $g$ over every finite orbit is zero, though it requires a clever trick. This is a result of Livsic.)