We demonstrate that the three-dimensional (3D) binary monoclinic oxides HfO2 and ZrO2 exhibit quasi-2D polaron localization and conductivity, which results from a small difference in the coordination of two oxygen sublattices in these materials. The transition between a 2D large polaron into a zero-dimensional small polaron state requires overcoming a small energetic barrier. These results demonstrate how a small asymmetry in the lattice structure can determine the qualitative character of polaron localization and significantly broaden the realm of quasi-2D polaron systems.