Abstract: The elastic scattering between dark matter particles and radiation represents an attractive possibility to solve a number of discrepancies between observations and standard cold dark matter predictions, as the induced collisional damping would imply a suppression of small-scale structures. We consider this scenario and confront it with measurements of the ionization history of the Universe at several redshifts and with recent estimates of the counts of Milky Way satellite galaxies. We derive a conservative upper bound on the dark matter photon elastic scattering cross section of sigma gamma DM < 8 x 10(-10) sigma(T) (m(DM)/GeV) at 95% CL, about one order of magnitude tighter than previous constraints from satellite number counts. Due to the strong degeneracies with astrophysical parameters, the bound on the dark matter-photon scattering cross section derived here is driven by the estimate of the number of Milky Way satellite galaxies. Finally, we also argue that future 21 cm probes could help in disentangling among possible non-cold dark matter candidates, such as interacting and warm dark matter scenarios. Let us emphasize that bounds of similar magnitude to the ones obtained here could be also derived for models with dark matter-neutrino interactions and would be as constraining as the tightest limits on such scenarios.