= Solution
The restriction maps of this point sheaf satisfy the presheaf composition identities: every possible restriction is either the identity on $\mathbb C$ or the unique map to the zero group. We check the sheaf identity and gluing axioms for any open cover $U=\bigcup_iU_i$.
If $p\notin U$, every section group in the cover is zero, so a compatible family has the unique zero gluing. If $p\in U$, at least one $U_i$ contains $p$. For any two members containing $p$, their intersection also contains $p$, and compatibility forces their complex values to be equal, since the restriction maps there are identities. Call the common value $c$. Members not containing $p$ have only the zero section. The section $c\in\mathbb C_p(U)$ restricts to every member of the family, proving existence of a gluing; its restriction to any member containing $p$ determines $c$, proving uniqueness.
This also proves local identity for sections, and $\mathbb C_p(\varnothing)=0$ handles the empty open set. Hence
$$
\boxed{\mathbb C_p\text{ is a sheaf of abelian groups}.}
$$
It is the point-pushforward sheaf, usually called a <skyscraper sheaf>. Its restrictions are all surjective, so it is also a <flasque sheaf>. No separation assumption on $X$ is needed for the preceding sheaf-axiom argument.
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