Let $F$ and $G$ be sheaves over a scheme $X$. Assume we have a canonical morphism $ F\to G $ and, further, we have canonical isomorphisms at every fiber $F_{\mid x}\overset{\sim}\to G_{\mid x}$ .
If the morphisms on the fibers are induced by the global morphism, then we can conclude that $F$ and $ G$ are isomorphic, but in my case it's not easy to check directly that this is the case.
Does the fact that everything is canonical imply automatically that the morphisms on the fibers are induced by the global one?
What does "canonical" mean? It's not actually a technical term. If you mean something like I mean in this paper, then under the hypotheses given there, the answer is yes just because the main theorem implies that the stalk maps are in fact induced by the global map.
Note that your hypotheses on the sheaves are lacking: the induced maps on fibers being isomorphisms doesn't imply global isomorphism unless the sheaves are coherent (i.e. locally given by finitely generated modules) and flat (so locally free) in which case Nakayama's lemma does the job. You really want to say stalks here.
If you are implicitly asking about "how can we prove that two sheaves are isomorphic?" and are struggling with the question of how to "glue" isomorphisms given on each stalk or fiber, then the answer is, forget about that and just work with the global map directly. Construct globally, prove locally: that's the way with sheaves.