Generally agency-issued tickets aren't flagged as "show card".

The issue usually is for 3rd party tickets bought on short notice. Person A buys a ticket for person B flying in a few days. There's a high chance of fraud, as A may have stolen the card number and the chargeback doesn't occur until after B has flown.

It's not a bank requirement per se to show the card, but the airline can be held responsible for a chargeback if it isn't.

So it's a little unusual that you just can't show you're the same person as the cardholder.