When people take an oath, (based on my understanding of Court protocol, but I am no expert
) the main point is to have them swear
by something they value. So whichever holy book they hold sacred, at the swearee's choice.
Traditionally a Bible, since traditionally most people are Christian. Jews would bring in a Chumash etc. because "Bible" is a somewhat ambiguous term. The Christian Bible (Old+New Testament) != Jewish Torah (which is NOT the Old Testament).
(This is entirely beside the point that the Christian Bible itself has multiple conflicting texts, so how do you know which one of them to believe in. 
)
Nowadays it is somewhat common to
"affirm" instead of swear, if you don't believe in G-d. Because people nowadays are G-dless heathens and suchlike, naturally.
