There was a period in the nineteenth century in which rather than, for the most part, obtaining paper money from the Treasury, banks were allowed each to print their own dollar bills. But the dollars each bank printed out weren't all accepted at the same rate, and no doubt regional variation in the dollar for dollar exchange rate also went into that.1 The number of dollar bills of one bank that another bank would accept would vary depending on the bank.
Moving from this to my discussion: If we equate the exchange of banknotes to the exchange of “receivables” in a system in which banks
can (either deal directly with each other or) exchange receivables for their own payables in a clearinghouse type arrangement, and
In so far as the two banks submit their mutual cross payments to such clearinghouse settlement-by-cancellation (across many, many banks, that is) and the payment settlement requirement is non-instantaneous enough to allow for intermediation of loanable funds of such receivables,
Then, the act of a bank with excess receivables “loaning” these out to the highest bidder during the allowed non-payment period2 (whether it be overnight or several days) can be seen as not dissimilar as to variation in rates received (assuming no “going” rate) to the paper bank bills example.
Selgin, in an interview I watched, suggests the variation in bank note exchange rates in the United States came from the purely local nature of most U.S. banks, as generally enforced by local laws aimed at preventing local competition from out of state or even out of town banks. The rates of a bank that crossed large distances among its branches would average out, while rubbing up also against the going rates in those distant cities.
https://www.actumprocessing.com/knowledge-base/101-on-automated-clearing-house-ach-and-settlement-period/
Payment processors and banks typically enforce a settlement period before releasing the funds into a merchant’s bank account. Typically, the period can be three (3) to five (5) business days.