The following are citations from the classic Applied and Computational Complex Analysis, Vol. I by P. Henrici. The chapter 3: Analytic Continuation provides a thorough treatise of the theme. Here we look at two aspects, which should help to clarify the situation.
At first we take a look when two functions $f(z)$ and $g(z)$ are analytic continuations from each other.
Theorem 3.2d: (Fundamental lemma on analytic continuation)
Let $Q$ be a set with point of accumulation $q$ and let $R$ and $S$ be two regions such that their intersection contains $Q$ and $q$ is connected. If $f$ is analytic on $R$, $g$ is analytic on $S$, and $f(z)=g(z)$ for $z\in Q$, then $f(z)=g(z)$ throughout $R\cap S$ and $f$ and $g$ are analytic continuations of each other.
We observe, we need at least a set $Q$ with an accumulation point where analytic functions $f$ and $g$ have to coincide. This set is part of the intersection of two regions $R$ and $S$ where $f$ and $g$ are defined. Finally we conclude that throughout $R\cap S$ the functions coincide.
The second aspect sheds some light at functional relationships in connection with analytic continuation. We can read in
Section 3.2.5: Analytic Continuation by Exploiting Functional Relationships
Occasionally an analytic continuation of a function $f$ can be obtained by making use of a special functional relationship satisfied by $f$. Naturally this method is restricted to those functions for which such relationships are known.
He continues with example 15 which seems that P. Henrici had precisely a user with OPs question in mind.
Let the function $g$ possess the following properties:
We assert that $g$ can be continued analytically into the whole complex plane with the exception of the points $z=0,-1,-2,\ldots$.
We first continue $g$ into $S:\Re (z)>-1,z\neq 0$. For $z\in S$,let $f$ be defined by
For $z\in S, \Re(z+1)>0$. Hence by virtue of (a) $f$ is analytic on $S$. In view of (b) $f$ agrees with $g$ on the set of $R$. Since $S$ is a region, $f$ represents the analytic continuation of $g$ from $R$ to $S$. We note that $f$ satisfies the functional relation $f(z+1)=zf(z)$ on the whole set $S$.
Denoting the extended function again by $g$, we may use the same method to continue $g$ analytically into the set $\Re(z)>-2,z\neq 0,-1$, and thus step by step into the region $z\neq 0,-1,-2,\ldots$.
Of course this example addresses the Gamma Function $\Gamma(z)$ which is treated in detail in chapter 8, vol. 2. He then continues with further methods of analytic continuation, such as the principle of continuous continuation and the symmetry principle.