The notion of the electric dipole polarizability density function of atoms and molecules has been considered. The current density induced by the time derivative of the electric field of monochromatic light allows for a new definition of the electric dipole polarizability density, which is translationally invariant. This translational invariance provides the physical meaning that was lacking in previous defined polarizability densities. The new polarizability density has been implemented at the TD-DFT level of theory. The origin independence has been proven in silico to hold regardless of the basis set size. Two emblematic molecules, i.e., CO and N2, which in many respects display similar electric response, have been studied in detail. The substantial differences, which have been highlighted in the topology of the parallel and perpendicular polarizability density tensor components of CO and N2, are grossly hidden by compensation, when integration is carried out to get the molecular properties.