I don't know why anyone would think this has anything to do with religion except the ignorant.
Swearing on the Bible is widely accepted in Western society as a way of saying "I'm telling the truth". The reasons why that is accepted are historically tied to religion, but the phrase itself bears little relationship to the "religion" associated with the book. In fact, as a follower of that book, I argue that the Bible actually tells us not to do that!!
You're reading far to much into the statement and misunderstanding the cartoon.