is that the language in the Constitution for the oath in Article II Sect. 1 also says "swear or affirm", meaning it was explicitly attempting to accommodate those forbidden to "swear" (e.g., usually attributed to Quakers) and sort of neutralize the religious aspect.
But in the case of what you are talking about, it's a ceremonial feature.
When Keith Ellison (now MN AG) was first sworn in for his first term in Congress almost 20 years ago, he caused head explosions for wanting to use a Qur'an for that purpose. HE ended up using one that was owned by Thomas Jefferson -
Thomas Jefferson's Copy of the Koran To Be Used in Congressional Swearing-in Ceremony
IMHO, it's like the flag pin, that has politicized enforcement. Federal employees are not "sworn in" with a bible. In fact, if you watch any hearings where witnesses are giving testimony, none of them are being "sworn in" using a bible (or anything else). 
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