The following assumes that the holography to which the OP refers is that which is studied in high energy thoery. Holography is not just a framework that relates
something (the RHS) on the region to something (the LHS) on its boundary
It is a framework for studying the equivalence of certain theories, one of which is defined in the bulk of some spacetime manifold with boundary, and the other of which is defined on its boundary. On one side of the equivalence, one has a theory of gravity. On the other side of the equivalence, one has a quantum field theory. In particular, in order to produce an example of holography, one needs to find two such theories, and one needs to show that the quantities that characterize the boundary theory (e.g. correlation functions in a quantum field theory) can be computed in terms of the quantities that characterize the bulk gravity theory, and vice versa.
Stoke's theorem is a mathematical fact about integrating differential forms on manifolds with boundary; it is not an equivalence between a theory of gravity and a quantum field theory. Therefore it would, in my opinion, be quite a terminological stretch to say that it is an example of holography.