AC acknowledges the OP paid for the ticket and agrees they owe him a refund.
They're hanging him up because he doesn't have a gift card number?
So if he'd paid with a credit card that he'd lost, had voided and replaced with a new card with a new number, he'd be expected to produce the original card number, maybe its expiration date and CVC, before a refund is paid?
There are people who would accept that as a business practice?