At centimeter wavelengths, the young stellar system T Tauri is known to be composed of two sources, the northern one associated with the optical star T Tau itself, and the southern one related to the infrared companion T Tau S. Here we reexamine the origin of the radio emission from these two components using archival 2 cm, 3.6 cm, and 6 cm VLA observations. The emission from the northern member is confirmed to be largely dominated by free-free radiation from an ionized wind, while the southern radio source is confirmed to consist of a compact component of magnetic origin, surrounded by an extended halo. Only moderately variable, the extended structure associated with the southern source is most likely the result of free-free radiation related to stellar winds. However, its flat spectral energy distribution, its extent, and the lack of variation of its size with the frequency of observation are incompatible with the classical picture of a fully ionized wind with constant velocity and mass-loss rate leading to an electron density distribution of ne(r)~r-2. Instead, we propose a model in which the ionization results from the impact of a supersonic wind driven by T Tau Sb onto dense surrounding material, possibly associated with the circumbinary disk recently identified around the T Tau Sa/T Tau Sb pair. The timescales for cooling and recombination in such a situation are in good agreement with the observed morphological changes undergone by the extended structure as its driving source moves through the environment.