Of course you have a choice. I have a bunch of store discount cards in the name of:
Guy Fawkes
666 Gropekent Lane
London W1 England
(020)7234-5678
If I pay cash when using those cards, then good luck matching them with an ID.
I’ve actually paid cash for an air ticket before with no problem — on American Airlines, you can hold till midnight of the next day without paying. I reserved a ticket online the previous day, then actually bought it when I showed up for my 6 am flight at LaGuardia. No one searched my bags or even looked at me funny.
Generally, I’ll pay with a credit card because airlines other than AA won’t let you reserve the Internet price without inputting a card #.