Interestingly, you can say "a bible" if you're referring to one of many physical, printed versions of the book.
I have two bibles in my house, a bible from my grandmother's family, and a bible from my grandfather's.
Why is there a Bible in every hotel room?
When you say "the Bible," however, you're referring to the literary work itself, rather than a specific instance or printed version of the work. I think we simply think of it as a proper noun, as though "the" is part of the title.
Looking into this I did find this discussion, which may be of interest. It didn't seem to come to a consensus as to why this is grammatically, but some users mentioned the etymology of the word as likely contributing to this usage. "Bible" comes Latin biblia meaning simply "book." Essentially, people used to simply refer to it as "the book," and I suppose the definite article just stuck around while "Bible" evolved into a word of its own.
Worth noting is that there are a lot of historical literary works whose names use the, including religious texts (The Torah, The Qur'an) and epic poems (The Iliad, The Aeneid).