Trump not putting his hand on the Bible might not be anything sinister or have a hidden meaning.
Yeah, as I said, he could have just been distracted or "zoned out" and simply forgotten, though you'd think Melania, whose job it appeared to be to hold the Bible(s) [why two?], might have given him a reminder hint.
If that were me being sworn in, I would bring a Clemetine Vulgate.

... but then again, I'd have a difficult time swearing (on a Bible or just in general) to uphold this (Masonic) Constituion.