In the merchant processing industry, there are frontends (where your transaction initially is routed for approval by Visa/MasterCard/American Express/Discover or by one of the Debit Networks) and backends (where the amounts of these transactions are debited between banks).

Square and Stripe are both Independent Sales Organizations riding atop Chase Paymentech and First Data respectively: https://www.quora.com/What-payment-processor-do-Braintree-St...

Stripe does use Wells Fargo for the backend settlement, they are one of the most popular acquiring banks on First Data (though many other banks can be used for underwriting risk and settlement of payments on First Data).

Beware, Gravity screwed over numerous businesses I work with. They purposefully misclassified businesses SICs (a code that defines what very specific category your business is, eg: a bar or a restaurant) to reduce credit card processing fees (interchange fees) for their customers, but were caught doing this for a massive number of businesses.

The card networks and issuing banks that had this money stolen from them went after the businesses that dodged these fees illicitly, with bills of tens of thousands or hundreds of thousands of dollars for fees that should have been paid.