$\DeclareMathOperator{\ab}{Ab}\DeclareMathOperator{\qcoh}{QCoh}$ This entry in the nlab shows that for $A$ a (commutative unital) ring, the category $\mathsf{Mod}_A$ of $A$-modules is equivalent to the category $\ab(\mathsf{CRing}/A)$ of abelian group objects in the slice category of rings over $A$. A module $M$ is associated with the square-zero extension $A\oplus M$, with the familiar $$ (a,m) (b,n) = (a b, a n + b m) $$ Kahler differentials, and the cotangent complex are easily defined in this setting, and this question asks about geometric intuition.

Of course, the functor $M\mapsto \mathcal{O}_X \oplus M$ works for quasi-coherent modules on an arbitrary scheme $X$, so we get an embedding $\qcoh(X)\hookrightarrow \ab(\mathsf{Sch}^{op}/X)$.

My question is this: is this an equivalence of categories? If so, does the definition of Kahler differentials at the nlab recover $\Omega^1$? More generally, does this work for stacks?