I always thought Agnelli wore the original Brooks "Polo" button down (see Roetzel, Gentleman, at page 65). Another picture of Agnelli and watch over shirt cuff appears on page 227-the denim sportshirt Agnelli wears doesn't have a double cuff.
I also believe that Agnelli notoriously preferred BB shirts. I don't even know if Interno's been around long enough to have been making special hypoallergenic watch cuffs for Gianni -- I'd wager they weren't when he began doing it. The piece in esquire is simply repeating or paraphrasing what Interno and earlier puff pieces have said. I'll believe Agnelli had a skin condition when his doctor tells me; I also can't believe he'd wear as spivvy a shirt as Interno makes, which essentially appears to be for guidos who shop in Mayfair. I mean, as an Italian magnate, wouldn't he have worn a bit of metal jewelry -- a chain, rings, a bracelet, whatever? Is Interno going to market the shirt with a special channel for a gold chain to avoid the skin next?