I had never heard of Swish until now. Very interesting. I know there are some differences, but how is it fundamentally different from something like Square? https://squareup.com/
Not familiar with Square but Swish started as an app for payment between individuals, such as friends sharing a restaurant bill or payment of used things (think Craigslist etc). People even use it as payment for used cars. It was developed by all major banks in cooperation, so huge network effect from start, uses the well established "mobile bank id" for authorization and is instant and currently free. The instantaneousness means that you feel safe to hand over the keys (in the car sale case) the moment your app beeps to confirm the transfer, which is within a few seconds after the buyer authorized it on his phone.
Some additional info: your phone number is identifying the transfer target, and is associated with your debit bank account number behind. So you don't have to share anything additional, if your friend or the buyer has your phone number he can initiate a transaction.
Square has a mobile phone payment option, but most of their actual use is as a small-merchant credit card processor. They have no integration into larger stores (which have their own card readers), and as a result there's no real point for users in setting up their mobile pay system. My credit card works with Square and everything else, an app would only work with Square.
Later a version for merchants was developed.