It is not true, but something similar is true. If $X$ is a scheme, the functor you describe is actually $\mathsf{Qcoh}(X) \to \mathsf{Ab}(\mathsf{QAlg}(X)/\mathcal{O}_X)$, where $\mathsf{QAlg}(X)$ denotes the category of quasi-coherent algebras on $X$, and $\mathsf{QAlg}(X)/\mathcal{O}_X$ is the slice category consisting of homomorphisms $A \to \mathcal{O}_X$ (which is anti-equivalent to the category of affine morphisms $Y \to X$ together with a section $X \to Y$; it is not just $\mathsf{Sch}^{\mathrm{op}}/X$!). It is equivalent to the category of non-unital quasi-coherent $\mathcal{O}_X$-algebras (i.e. semigroup objects in $\mathsf{Qcoh}(X)$), by taking the kernel of $A \to \mathcal{O}_X$. Those non-unital qc algebras $B$ for which the addition map $B \times B \to B, (a,b) \mapsto a+b$ is a homomorphism are precisely those with trivial multiplication, i.e. which are just qc modules. And this is the only map which could make $B$ an abelian group object.

Thus, the same proof as in the affine case works. Besides, we don't really use schemes or quasi-coherence here: If $X$ is an arbitrary ringed space, then $\mathsf{Mod}(X) \cong \mathsf{Ab}(\mathsf{Alg}(X)/\mathcal{O}_X)$.

The connection with Kahler differentials is fine: If $M$ is some module on a ringed space $X$, then homomorphisms $\mathcal{O}_X \to \mathcal{O}_X \oplus M$ in $\mathsf{Alg}(X)/\mathcal{O}_X$ correspond 1:1 to derivations $\mathcal{O}_X \to M$.

Finally a remark about the nlab, which I couldn't resist to include here. In my opinion, the statements at the nlab about "**The** correct definition of **the** notion of module ..." and "... **the** correct definition of derivations and Kähler modules" should not be taken too seriously. This is a quite subjective point of view, which may explain some aspects for modules quite elegantly, but not all of them. Besides, there are lots of notions of modules, let alone modules for monads.