The most important observation for dark matter, really, is the cosmic microwave background. Basically, before the CMB was emitted, the universe was a plasma. Normal matter in a plasma interacts very strongly with light, and when it falls into a gravitational potential well, this interaction causes the normal matter to feel pressure and bounce. Dark matter, on the other hand, doesn't interact with light, and so experiences no such bounce.
Observing the relationship between how much matter falls into a gravitational potential well and how much bounces gives us a very accurate measurement of the ratio between normal matter and dark matter.
Your "normal matter in an alternate universe" would bounce just like normal matter here does, so it would show a dramatically different signature in the CMB.