Every pedal brand says to use only their power supply, because (a) they make money selling power supplies, and (b) it avoids a legal liability problem.
The fact is brand has nothing to do with it, and techie jargon has everything to do with it. You have to learn and understand what specific parameters of power are needed, and what is actually provided by each supply.
Conveniently, most pedals accept the exact same power supply as each other. Boss, Dunlop, and Ashdown all take the same power and can use each other's supplies.
But just because that's the convenient majority does not let you off the hook from learning how they all work.