We have good reason to believe that when we talk about dark matter, we really are referring to some kind of "stuff" that has mass and momentum, but doesn't interact electromagnetically with normal matter.
The most visually striking evidence comes from the Bullet Cluster. We can see two galaxy clusters that have recently collided at ~1% of the speed of light, causing the gas to be slowed and heated. And we can map the mass distribution via gravitational lensing, and see that a large amount of the clusters' mass has passed right through each other without being slowed. This is more or less what you would predict if "dark matter" is made of weakly-interacting massive particles (WIMPs), and is pretty hard to explain by appealing to a modified law of gravity.