As it is stated, the formula is not true (even up to some normalisation factor). Consider a connection $d+\omega$ on a complex line bundle, where $\omega\in\Omega^1(M,\mathbb C).$ The curvature is just $d\omega$, and the parallel transport along a curve $\gamma$ is just $exp(-\int_\gamma\omega)$. Thus, if you can apply Stokes theorem you can derive the holonomy formula. But this is not always the case, as you see from the example of the punctured disc (with boundary $S^1$) and the flat connection $d+a\frac{dz}{z}.$

As it is stated, the formula is not true (even up to some normalisation factor). Consider a connection $d+\omega$ on a complex line bundle, where $\omega\in\Omega^1(M,\mathbb C).$ The curvature is just $d\omega$, and the parallel transport along a curve $\gamma$ is just $exp(-\int_\gamma\omega)$. Thus, if you can apply Stokes theorem you can derive the holonomy formula. But this is not always the case, as you see from the example of the punctured disc (with boundary $S^1$) and the flat connection $d+a\frac{dz}{z}.$

There is no direct generalisation to non-abelian Lie groups $G$. For example, there exists flat $SU(2)$-connections on the 1-holed torus with non-trivial monodromy along the boundary.