The librarian analogy is a little stretched, but I do expect librarians to properly put fiction books in the fiction section and nonfiction books in the nonfiction section.
"Non-fiction has numbers", we were told.
I'm not sure Dewey attempts to separate fact from fiction -- just to categorize the author's intent (or assertion).
In addition to the Christian Bible, the "non-fiction" section also contains religious texts that refute it. And New Age crystal vibration harmonic ecstasy stuff. And Dianetics.
The wide disagreement on "fiction" vs "non-fiction" is probably not a solvable problem. See also: a world map, at any time in recorded history.
That's not to say the line isn't murky, but that's true of any human classification system.
Speak for yourself. Several atheists including myself read some of the Bible for our book club. In particular the wisdom literature has value as a fiction removed from the religious context. Beyond that, the Bible is the earliest work we have which exhibits several tropes, and western art has been heavily influenced by it.
Fictional works are organized alphabetically by the author's last name
It's pretty clear that the anthology/compendium often termed "The Bible" is mostly myth and retellings of oral histories that date back many millenia, with some of those incorporating historical events (think docudrama[0]).
As such, I'd posit that in a vacuum, such works should be placed in the "Historical Fiction" section.
However, since there are quite a few of these works (the Hindu Vedas, the Tripitaka, the Quran, etc.), it seems that the current classification (Religion) is reasonable.
That said, not specifically calling it "fiction" doesn't make it non-fiction.