This question is coming from physics, so I hope you will be patient as you read the following.
Setup: Assume all criteria for applying Stokes' theorem are met. Consider a 2D plane representation of a torus, where the left-right and top-bottom edges are identified with each other. On this plane, I choose a closed counterclockwise circular loop. Per my physics context, this loop defines the boundary of a submanifold on the torus. By convention, the area of integration for Stokes' theorem in this case would be the area inside the loop; and the area inside + the circular loop are respectively the region and boundary of the submanifold in consideration. The focus is on this submanifold, and not the common behavior of Stokes' on the entire boundary-less torus as discussed in other questions. Anyway, Stokes' theorem would then relate the line integral along this loop to the area integral of the region inside it.
Question: The above case of a circular loop is clear to me. However, one may also choose a circular loop on the torus by choosing either a vertical line or a horizontal line on the 2D plane. These lines are circular loops as well because the left-right and top-bottom edges are identified with each other. However, it is not clear to me how to apply the notion
As people have commented, there is no Stokes's Theorem application here. What there is is a notion of periods (analogous to Gauss's Theorem with point charges in electrostatics for integrating over closed surfaces). If you have a closed $1$-form $\omega$ on the torus, its integrals over the "vertical" circle and the "horizontal" circle uniquely determine the cohomology class of $\omega$ (i.e., determine $\omega$ up to the addition of an exact $1$-form — corresponding to a conservative force field in physics terms).