If it's purely gravitational then that implies infinite density and infinite force for any matter that occupies the exact same space.
But I will briefly point out the flaws with your thought experiment:
1. Point-like particles don't exist, so the force will never become infinite. In fact, this is the wrong paradigm all together -- if you're talking about two particles "colliding", then what you're really referring to is their interaction cross section (which you can think of as being their probability of interacting). Neutrino (weak interaction) scattering is incredibly rare. Gravity is much weaker still. The particles would indeed pass right through each other then oscillate back and forth (not forever, but practically so).
2. You have assumed that both particles begin with a relative velocity of 0. This is not a good assumption. In the real world, two dark matter particles would have random relative velocities according to some distribution, which means they would have angular momentum relative to each other. Again, that angular momentum can only dissipate through friction. If the particles interact solely gravitationally, then they can only lose energy via gravitational waves (i.e., extremely slowly).
You probably know that the Earth formed due to gravity. But gravity is only half the answer -- you also need something to slow the matter. On Earth, that was the electromagnetic force. Purely gravitational matter has only gravity.
Because DM passes right through itself it doesn't clump like the stuff that made our planets do. Instead it remains spread out bound by gravity but not clumping. That makes it incredibly diffuse. But space is also quite big, so it can still contribute a lot of mass.
