So, Felix picked up a white pawn, and set it down on its original spot, on his chess set that has three white bishops and nine white pawns... what could possibly be the meaning behind this...?
Actually, I shouldn't be too harsh. Some assistant somewhere may have goofed up filling in the pawn colors from the reference, but the actual position is an ancient, well-known one that arises out of the Italian. The black pieces in disguise are the bishop on g4 (pinning the white knight to the queen) and the pawn on e5 (the one of the two central pawns closer to Black).
As for Felix idly touching the pawn on a2 for dramatic effect, while pondering how to properly utilize his new-found human "pawn", one Monica Norton (very subtle)... Either pawn to a3 or pawn to a4 allows 5...Nd4, putting additional pressure on the pinned knight, so he was right to reconsider. If he had picked up the rim pawn all the way on the other side, he might have played 5.h3, which is pretty much the best move in this position. If Black plays lazily and retreats the Bishop to h5 while upholding the pin, a cold shower awaits in the form of 6.Nxe5!!, disregarding the pin entirely. If Black takes the queen it's mate in two.
The position is pretty much ancient opening theory (to the point that the aforementioned mating sequence has a name: Legal's Mate), so it makes quite a lot of sense for someone who owns a chess set for study purposes to have it set up by chance. The research done here is not bad at all for what is intended as a mere prop for a smarmy princeling to fondle as he smarms. If it wasn't for those two pieces colored in wrong...
Yes, I did watch the anime and read the light novel and am now reading the manga specifically to see how the manga adaptation handled the bits with chess in them. How did you know? (Also, the anime did a really good job, I thought, so I wanted to see how much of the legwork had been done by the manga adaptation already.)