A button collared shirt was originally sportswear and it is usually made from a more casual cloth like oxford or chambray. The advantage of the style is that it keeps the collar in place when worn under a sweater or without a necktie under an odd jacket.
In my opinion most of Agnelli's expressions, like the watch worn over the shirt sleeve or pairing a buttondown collar with a double breasted suit, struck the viewer as unusual rather than unattractive. Flapping collars on the other hand look awkward.
Perhaps wearing them is simply an easy form of sprezzatura to remember in the morning.