Polarization control with nanostructures having a tunable design and allowing inexpensive large-scale fabrication is important for many nanophotonic applications. For this purpose, we developed and experimentally demonstrated nanostructured plasmonic surfaces based on hexagonal arrays of anisotropic coaxial nanocavities, which can be fabricated by a low-cost self-assembled nanosphere lithography method. Their high polarization sensitivity is achieved by engineering anisotropy of the coaxial nanocavities, while the optical response is enhanced by the excitation of surface plasmon resonances. Particularly, varying the geometrical parameters of the coaxial nanocavities, namely the height and tilt angle of their central core nanoellipsoids, the plasmonic resonance wavelengths as well as the polarization-selective behavior can be individually tuned in the entire visible and near-infrared spectral regions, which makes such nanostructures good candidates for the implementation of polarization-controlled optical switches and polarization-tunable filters. Moreover, the developed nanostructures demonstrate sensitivity up to 1335 nm/RIU in refractive index sensing.