Why thermodynamic state variables of a sytem are only defined at thermodynamic equilibrium and not during any other instant?
Variables such as T and p are not well-defined when a system is fluctuating. Thermodynamic state variables are defined for systems that are static or quasi-static (in the latter case, transitioning very slowly - and therefore reversibly - through a series of equilibrium points). The main reason is that macroscopic fluctuations are absent in true equilibrium states. Each phase in such states is homogeneous. The argument is thus somewhat circular: thermodynamic states are homogeneous states that have fixed properties. The only way to guarantee fixed properties is by being at equilibrium.

Note that it is possible to obtain thermodynamic parameters by imposing restraints that trap a system in a local free energy minimum, and thereby prevent the system from transitioning to a global free energy minimum.