When a card is swiped the "merchant acquirer's" bank/card processor is contacted to effect clearing settlement.
Square is the "merchant" in the traditional sense of how a card swipe transaction transaction occurs. Square can do whatever they want with the proceeds. They just happen to, today, sweep those proceeds (less 2.75%) to the Square dongle account-holder's bank account. If they instead wanted to let their account-holder's accumulate a balance for all card-swipe payments received, they could do that.
I cannot yet imagine how Square is not just another interface into the Visa/MC networks.
Today, that's all they are. But they aren't making much money because they charge just a 2.75% swipe fee and are in-turn paying most of that out in fees as well.
But once Square adds the concept of a wallet, then if a Square user (Alice) does a $100 sale she gets $97.25 but she lets it sit in her Square wallet. Then when she uses it to send money to another Square user (Bob) , Square gives Bob just $94.57 ($97.25 less 2.75%) and keeps the $2.68 difference. That' how PayPal has been functioning since 2001. There's nothing tricky about it.