The frontend (what authorizes the transaction with the card network, like Visa, Pulse, Star, Mastercard, etc) and backend (settlement bank that ACHes the cash around and usually takes the underwriting risk (unless the independent sales organization, eg: Stripe or Gravity decides to do this themselves)) are generally called a payment platform in the industry.
This platform can get wrapped with many different payment terminals (Verifone, Pax, Dejavoo, Ingenico, etc) or processing interfaces (Authorize.net, PayPal, Stripe, Square, et all).
Square does look to be a bank now, but it appears the only products they are underwriting risk on are loans to businesses, which they immediately try to sell off to third party investors: https://www.cnn.com/2021/03/02/tech/square-bank-business-len...
Kinda surprising, I would expect them to have underwritten their own payment processing, but it might be the case that they don't want to have the outsized liability of chargebacks on their books if a client business goes bankrupt, all to save 2 cents on a $100 transaction.
Perhaps once the US economy slows its rapid economic shift they will consider handling their own underwriting and batch settlement.
