In the US, there's nothing requiring a religious book. Or even a book at all. While most people use the Bible (or their own religious texts), many have used other objects. Law books (John Q Adams) have been used. Teddy Roosevelt used nothing. In 2017, Lan Diep took his oath on a replica of the Captain America shield.

The law also doesn't (usually) specify who can administer the oath. What matters are the words spoken.