What does one need to assume in a category $\mathcal{C}$ with an initial object $0$, so that every morphism $f:0\to A$ out of $0$ (I am not assuming anything on the object $A$) satisfies the property that:
whenever $f$ is a monomorphism it is necessarily an isomorphism ?
Are there any natural examples of such categories?
If every monomorphism out an initial object $0$ is an isomorphism, then every object is (canonically) isomorphic to $0$, i.e. the category is an indiscrete category, i.e. two objects have exactly one morphism between them.
To see this, first consider an initial object $0$ every monomorphism out of which is an isomorphism. Then $0$ is a strict initial object, i.e. any morphism $A\to 0$ is an isomorphism. Indeed, for any morphism $A\to 0$, the composite $0\to A\to 0$ is always the identity morphism by universal property of initial object. Consequently, $0\to A$ is a (split) monomorphism out the initial object, hence by assumption an isomorphism. But then $A\to 0$ is a right inverse of an isomorphism, hence is itself an isomorphism.
Second, observe that if $0$ is a strict initial object in a category, then it is also subterminal: any morphism from an object to $0$ is unique. Indeed, given $B\underset g{\overset f\rightrightarrows} 0$, since $f$ and $g$ are both isomorphisms, we must have that $0\xrightarrow{f^{-1}}B\xrightarrow{g} 0$ is the identity morphism, hence $g$ is the inverse of $f^{-1}$, i.e. $f$.
Consequently, in a category with a strict initial object $0$, every morphism out of $0$ is (vacuously) a monomorphism since any pair of morphisms $B\rightrightarrows 0$ is a pair of identical morphisms. Thus, if every monomorphism out of an initial object is an isomorphism, then every morphism out of $0$ is an isomorphism. Hence, an initial object has a unique isomorphism with every other object, so all objects are initial, so any two objects have exactly one morphism between them.