If God gives those experiences to people of all religions, then those experiences are no longer evidence for any particular religion.
It simply suggests that the criteria that God uses, when deciding when to make His presence known, are not limited to the professed religion of the person in question. Exactly what those criteria are, is not fully clear; but any religion which helped to foster those criteria amongst its adherents would be at least partially correct, in its effects if not necessarily in its dogma.
I assume you believe that the Devil can't hand out mystical experiences
Why not? We're talking about the what may be second-most powerful entity in existence. (Mind you, there's a big difference between can and will; the existence of the Devil strongly implies the existence of God, and if the Devil is trying to discourage religion, then it would be counterproductive unless some other effects of said experience outweigh that risk.)
It simply suggests that the criteria that God uses, when deciding when to make His presence known, are not limited to the professed religion of the person in question.
That sounds like you're agreeing with me. If God gives people mystical experiences no matter what their religion, mystical experiences are no good in showing that the religion is true. Of course this is a matter of degree. The wider the range of people get the experiences, the worse the experiences are at showing anything.
If beings other than God (such as the Devil) can give out mystica...
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.