I would totally expect them to say the same thing about their books and I would absolutely agree. The problem isn't with the claim but with the inconsistency of many that accept let's say...the Quaran as the authority on Allah but refuse to accept the Bible as the authority on God.
What evidence...scripture tells us that the evidence that God has given us as to our salvation is the HS. The evidence of that HS is a changed life. It is a desire and goal of righteousness, it is Love, Joy, Peace, etc. Galatians 5, the fruit of the Spirit for those that might want the full list. When these things are present and in the form that the Bible tells us is consistent with the HS, we have tangible evidence of our salvation. It may not be measurable as in X number of inches or Y ounces, but it is measurable in light of who we were before and who society is at large. Without that evidence, we simply can't be sure if it is your religious belief or your salvation.
How do you know that you are believing in accordance with reality, while they are obviously incorrect? Test. The Bible tells us to test the spirits, to test things, not just accept them because it is what you were taught. I have been a believer for almost 50 years now. Over that 50 years I have taken very serious the admonition of the Bible to test and interesting enough, I have yet to find one claim that is actually made in the bible to be false. Some haven't yet happened, some we can't be sure, but of all those left, I haven't come across one false one yet. But this also goes back to the claim that the Bible is the authoritative word of God. You see, this is where many people get all crazy. They listen to teachers of God tell them what the Bible says and never look at it themselves. Just had a discussion about this yesterday over the topic of overcoming. The Bible tells us that in Christ we are more than overcomers (one way to measure) yet when I brought that up to a "christian" that is one who claims belief but lives like an unbeliever, they were upset to even think that anyone (refused to even acknowledge scripture says it) would suggest not allowing our abuse to define us was a good thing much less do able. Yet I know of people who live above their abuse and I am one of those people. Evidence of the Living God of the Bible. Take all that evidence and add it together and you have a very strong case for knowing truth from delusion. that would be why we should test them to see.Because other people see it too...when I first came to Christ, I prayed that if there was truth to be had, people would notice a difference without me even saying anything. Low and behold, without me saying a single word, people started noticing a difference. This is one of many measurable ways to know truth from delusion when it comes to what one believes.