Aren't they referring to their credit card?
Carta Moneta is a kind of credit card issued by several Italian banks (which are part of the INTESA group) and it allows you to spend money through this system called "moneta", either through VISA or MASTERCARD circuit.
When you're at the cash and you give them your credit card, they might ask: "bancomat o moneta?" Bancomat is when they charge you immediately, moneta is when the money will be debited at the end of the month or so, depending on the bank conditions.
To know more:
http://www.bancaintesa.it/piu/jsp/Home
Or maybe you were not talking about Italian tourists in Australia... who knows?