This is true for sets by the Schröder–Bernstein theorem where monomorphisms are injective functions. Is there a general type of category in which this condition holds? I suspect it holds regular categories or maybe abelian categories. (I will accept any reasonable answer more general than just the category of sets.)
This is not true in general; consider the category freely generated by $f:A \to B$ and $g:B \to A$ in which $f$ and $g$ are both mono (somewhat vacuously) but are not isomorphic.
My attempt was to start with abelian categories since it is the strongest (and work backwards): assume $f$ and $g$ are both normal. Then they are the kernel of some $\alpha:A \to C$ and $\beta:B \to D$ with $f\beta:A \to D$ and $g\alpha:B\to C$ being zero morphisms (writing composition from left to right). So we can multiply on the left freely:
$$ f\beta = fgf\beta=fgfgf\beta=\cdots $$
$$ g\alpha = gfg\alpha =gfgfg\alpha=\cdots$$
By the universal property of the kernel we can choose $k=fgf$ so that $k\beta$ is a zero morphism and find unique $u:A \to A$ such that $uf=fgf$. Since $f$ is mono we have $u=fg$. (I was hoping $u$ would be the identity.)
Likewise we can take $k'=gfg$ so that $k'\alpha$ is a zero morphism and find $u':B \to B$ such that $u'g=gfg$, so that $u'=gf$. Again, not making much progress...