I know that nn an abelian category $\mathcal A$ we have the following property:
$$\require{AMScd} \begin{CD} 0 @>>> A@>f>> B @>g>> C @>>> 0 \\ @. @V\varphi VV @V\psi VV @| @. \\ 0 @>>> A^\prime @>f^\prime>> B^\prime @>g^\prime>> C @>>> 0 \\ \end{CD} $$
If both rows of the commutative diagram given above are exact, then $\{\>f^\prime,\psi\}$ must be a push-out of $\{\varphi,f\}$.
I'm just curious broad just how generally this property will apply, that is, is there some class of categories broader than that of abelian categories such that the above property still hold within it?
Semi-abelian categories include all abelian categories, but also groups, non-unital rings, Lie and Leibniz algebras over any ring, cocommutative Hopf algebras over a field of characteristic zero, (pre)crossed modules...
They have a lot of nice properties, including the one you give here. It appears as Proposition 1.5 of the paper "On the second cohomology group in semi-abelian categories" by Gran and Van der Linden (but it may have appeared earlier).
On the other hand, the dual property is valid in homological (and even sequentiable) categories ; so your property holds in any "co-homological" category.