I'm not aware of any theoretical requirement for dark matter to be massless in order to be "collisionless," but since I am neither a particle physicist nor play one on TV, I'm pinging someone who is, in case he can shed more light on the subject.
While we're waiting for the pros to turn up, another amateur opinion.
Dark matter has to have some kind of effective mass or it wouldn't be invoked to explain what it explains. It just doesn't have a gas pressure. Gas pressure (the effect of particles colliding) works against clumping. If particles that had gas pressure were all there is, space would look a certain way.
Having some of the mass in a space unable to bump into other particles, even its own kind of particles, helps stuff (including the stuff that does have collisions) clump up gravitationally better. Space looks a little different.
Which is why dark matter was hypothesized in the first place. The gravity wells of galaxies and eventually other spaces were deeper than we could explain without something invisible and very different from normal matter made of protons and neutrons.
Meanwhile, most of the mass-energy of the universe is something called "dark energy." That's a different can of worms yet.