The gluing axiom states that given an open cover $\{U_i\}$ of some open set $U$, if I have a section $s_i$ on each $U_i$ such that the restrictions agree on intersections, then there exists a unique section of $U$ such that its restriction to each $U_i$ is $s_i$.
Does dropping "unique" from the gluing axiom lead to anything mathematically interesting?
I think a trivial example of this sort of sheaf is a topological space $X$ where $F(U)$ for any proper subset of $X$ is the trivial group, while $F(X)$ is any nontrivial group. Then if I take an open cover of $s$, I can glue all of those trivial sections into any element of $F(X)$.