It is well-known the notion of weak limit in category theory. To obtain it, just take the notion of a limit and drop out the uniqueness property in the universal property.
Now, as the universal property is crucial when proving the uniqueness (up to isomorphisms) of a limit, I expect the non-uniqueness of weak limits, in general. Well, it is possible to find an explicit example of category that admits weak pullbacks, where a cospan diagram admits two non-isomorphic weak pullbacks?
Consider a category $\mathscr{C}$ and a cospan $X\rightarrow Y\leftarrow Z$. Let $X \leftarrow L \rightarrow Y$ be a cone over the diagram. Dropping the uniqueness condition in the case of pullbacks amounts to saying that for any object $T$ the canonical map $\mathscr{C}(T,L) \rightarrow \mathscr{C}(T,X)\times_{\mathscr{C}(T,Y)}\mathscr{C}(T,Z)$ is surjective (instead of bijective).
Taking $\mathscr{C}=\mathsf{Set}$, $X=Y=Z=T=\{\ast\}$ you find a plethora of inequivalent weak pullbacks.