They will often write your folio on paper in the MDR on the ship too, especially if you buy the shots in the souvenir shot glass (or the drink of the day on Lido). It's just faster for them at that instant.
When you purchase in the gift shops or at the bars on the island, all the orders are uploaded at the end of the day before the ship debarks. It isn't realtime communication with the shipboard S&S systems. So it is possible, horror of horrors, if they did do Cheers on the island they could serve you more than 15 drinks!