Well, in all honesty, personal testimony, stories in ancient texts (like the Bible, Quran), are evidence. They are the poorest, least reliable type of evidence there is.
The reason why they are the least reliable and poorest type there is, is because they are not testable, demonstrable, and falsifiable.
And yes, most theists do have a different standard of evidence for their particular religious beliefs, than they do for other poorly evidenced supernatural claims.
I am pretty sure, that the majority of theists disbelieve many other things in common with most atheists (bigfoot, alien abductions, hollow earth, Loch Ness monster, etc, etc). And they probably disbelieve then for the same reasons we do; lack of demonstrable evidence. They just don’t seem to apply the same standards of evidence for their god beliefs.