It’s actually not mandatory that a Bible, or any religious text be used for swearing in a president. There’s nothing stating that a Jewish president couldn’t use the Torah or a Muslim president couldn’t use the Koran. We’ve just only had Christian presidents so far, though not all of them have used bibles for the ceremony.

Separation from church and state only pretty much states that congress can make no laws favoring one religion over another or make any laws prohibiting the practice of one’s religion. To prohibit a president from swearing in on a religious text of their choice would, in and of itself, be a first amendment violation. Saying they have to, would also be a violation. The strict separation of church from the state, freedom from religion or the “wall of separation,” is something people have argued for, but isn’t actually laid out in the constitution.