If the religion you were born into determines the religion of the book(s) you write, wouldn't that mean Buddhism is actually just Hinduism, since its earliest followers (including Siddhartha Gautama) were Hindus?
They may have been Jewish at birth, but they'd come to Christ's teachings since then. As a Christian is a follower of Christ, I would argue that it was Christians writing them. But I guess that's an argument of semantics.