We study the properties of the dark matter component of the radially anisotropic stellar population recently identified in the Gaia data, using magneto-hydrodynamical simulations of Milky Way-like halos from the Auriga project. We identify 10 simulated galaxies that approximately match the rotation curve and stellar mass of the Milky Way. Four of these have an anisotropic stellar population reminiscent of the Gaia structure. We find an anti-correlation between the dark matter mass fraction of this population in the Solar neighbourhood and its orbital anisotropy. We estimate the local dark matter density and velocity distribution for halos with and without the anisotropic stellar population, and use them to simulate the signals expected in future xenon and germanium direct detection experiments. We find that a generalized Maxwellian distribution fits the dark matter halo integrals of the Milky Way-like halos containing the radially anisotropic stellar population. For dark matter particle masses below approximately 10 GeV, direct detection exclusion limits for the simulated halos with the anisotropic stellar population show a mild shift towards smaller masses compared to the commonly adopted Standard Halo Model.