In the English translation of Serre's FAC (link to the PDF can be found in this mathoverflow discussion in the top answer) Serre gives his first example of a sheaf of abelian groups, the constant sheaf, on page 8 in section 1: Operations on Sheaves. The following is verbatim from the translation, however do note that Serre defines a sheaf from the étalé space point of view, the definition is also given on page 8 or can be found online.
Example of a sheaf. For $G$ an abelian group, set $\mathscr{F}_x = G$ for all $x \in X$; the set $\mathscr{F}$ can be identified with the product $X \times G$ and, if it is equipped with the product topology of the topology of $X$ by the discrete topology on G, one obtains a sheaf, called the constant sheaf isomorphic with $G$, and often identified with $G$.
For the most part I understand this example. I am able to verify that the group operation maps are continuous, and the projection map is a local homeomorphism, so $\mathscr{F}$ is indeed a sheaf using the definition that Serre gives. Then for the sake of exploration I studied the sections of this sheaf and found that $\Gamma(U,\mathscr{F})$ is the group of locally constant functions $s\colon U \to \mathscr{F}$. Further, if $U$ is connected they become constant functions and then $\Gamma(U,\mathscr{F}) \cong G$, but if not connected $\Gamma(U,\mathscr{F}) \cong \oplus \, G$ where you get a summand for each connected component of $U$. None of this stuff is too exciting, and can be found on the wikipedia page for the constant sheaf. But what I do not get is
In what sense would this constant sheaf $\mathscr{F}$ be isomorphic to $G$? Why would we want to identify it with $G$? I see no way that the sheaf itself acts like $G$, and really it is more like no matter where you look on this sheaf (stalks, sections) you are either going to find $G$, or find copies of $G$. I checked the french version and it is not a translation error. Can anyone provide some context or explanation for this?
Here is another example of where Serre uses this language in a way that is confusing to me:
Every sheaf of abelian groups can be considered a sheaf of $\mathbb{Z}$ - modules, $\mathbb{Z}$ being the constant sheaf isomorphic to the ring of integers.
Also, here is another (less important) question that occurred to me while typing my original question up that I don't think is quite worthy of its own post (but I'll move it if someone complains):
Does anything weird happen when the subset $U$ has countably many or uncountably many connected components? Usually we would want to switch to a direct product of $G$ but that seems not to be what we would want, since we wouldn't want to require the sections to be nontrivial on only finitely many components?