I have put mushrooms of different color and texture next to each other without problem.
In my experience corallimorpharia (mushrooms, ricordia, etc) generally are not aggressive, though on occasion they can eat a shrimp or fish, same as anemones. Depends on the sizes though.
You will run into issues with many other species and groups though. most LPS have stinging tentacles that can reach far, and will kill anything. Most SPS do not sting other corals, and will lose every battle against every coral, fortunately they grow pretty quickly, and most soft corals will are more subtle about their attacking of other coral, either slowly poisoning them or overgrowing them.
As to the specific why with mushrooms not stinging each other, or some coral stinging some other corals but not all, I'm sure there are specific chemical biomarkers that differentiate self from predator/prey/competitor, but I don't know the specifics.