I'm pretty sure each person chooses a document/book to swear on that is core to them. So most people in the US would choose the Bible because they identify as Christian, but if a Jewish person or Muslim person was sworn in they could choose the Torah or Quran. And a non-religuous person could choose anything that they could convinceably argue is important/core to their values.
Disclaimer: I did no research right now to confirm this but that's what I remember.
Reminds me of the SkiFree Yeti