What is the name of this relationship between two functions?

in KCG Chapter 8 is called On Conjugacy, that being your relation. It is usually too much to expect anything global; in section 8.5 he begins to consider this relation in formal power series in $\mathbb C.$ More important than you would think. If the three series involved all have positive radius of convergence, he emphasizes this by saying analytically conjugate.

I do not yet see that the same phrase is used, but I can recommend A History of Complex Dynamics by Daniel S. Alexander. Maybe he has a different name.

An example, from Milnor, Dynamics in One Complex Variable: Let $0$ be a fixpoint of $f,$ holomorphic in a neighborhood. Take $$ f(z) = \lambda z + a_2 z^2 + a_3 z^3 + \cdots $$ Koenigs Linearization: if $|\lambda| \neq 0,1$ there is a local holomorphic change of coordinate $\phi$ with $\phi(0)=0,$ such that $$ \phi f \phi^{-1}(w) = \lambda w $$