The "network" part of what they do could straightforwardly be done in a decentralized way using ordinary DNS (for the parties to discover the servers of the others) and TLS. When your card issuer is Example Bank then your "card number" would be 123@example.com and the merchant or their bank knows to contact the servers at example.com to initiate a transaction for that card.

It seems like the primary thing they actually do is currency conversion, which a) isn't strictly required (e.g. plenty of small US merchants might hardly notice if they only accepted payment in US dollars), and b) could just as easily be transparently provided as an optional service by either of the card issuing bank or the merchant bank for cardholders or merchants who expect a non-trivial number of foreign currency transactions.