Question: does the vector bundle generated by this sheaf on X∖Sing is isomorphic to the cotangent bundle on X∖Sing is ? If not what is the sheaf we should define on X to get the cotangent bundle on X∖Sing ?
To answer both questions: Kähler differentials do indeed coincide with the cotangent bundle of X∖Sing X, as long as we take the correct definition of Kähler differentials.
The correct definition does not amount to some explicit formula, but rather defines Kähler differentials as the left adjoint to the square-zero-extension functor from modules to algebras.
An important subtlety here is to choose the correct notion of algebras:
In algebraic geometry we take ordinary algebras, which give rise to ordinary derivations and ordinary Kähler differentials. These recover the cotangent bundle of a smooth algebraic variety.
In differential geometry we take C^∞-rings, which give rise to C^∞-derivations and C^∞-Kähler differentials. These recover the smooth cotangent bundle of a smooth manifold.
In complex geometry, we take entire functional calculus algebras (EFC-algebras), which give rise to EFC-derivations and EFC-Kähler differentials. These recover the holomorphic cotangent bundle of a holomorphic manifold.
For a reference that treats Kähler differentials in this manner, see, for example, Pridham's A differential graded model for derived analytic geometry, Definition 3.16.