To contrast, Interac, the system used for canadian debit and person to person transfers, is a private company, with ownership by the banks.
If you look at their "about us" section, they just say they are "a corporation". https://www.payments.ca/about/who-we-are Although they were created by a law (Canada Payments Act) and have a public mandate:
Payments Canada has formed a critical part of Canada’s financial market infrastructure for over four decades. The Canadian Payments Association was created by the Canadian Payments Association Act in 1980, now known as the Canadian Payments Act (CP Act).
I always viewed it as one of the "we will govern and regulate ourself so government doesn't fully have to" bodies, but I'm an outsider so happy to be corrected :).