Drag is a complex phenomenon. In different cases, it's approximated as being linearly or quadratically dependent on velocity.
So if you think of just "the drag coefficient" -- you're thinking of an approximation for a certain object in a certain atmosphere in a certain range of speeds. Since changing mass will change terminal velocity, the best approximation of drag at terminal velocity may be different as well.
But, if you think of drag as a function of velocity, which may be linear or not, then, that function will be (almost) the same regardless of mass. So a ping pong ball traveling at $X$ m/s will feel the same drag force, regardless of mass.
I say "almost" because once you get to really low mass objects, I would expect that their vibrations and other movements in response to drag would start having secondary effects.