I was asked at Manchester by CX to see my credit card. Cash J ticket paid using an amex that I had cancelled months after booking and before travelling.
I ended up using the desk agent's supervisor's phone to log in to my amex account and show them the old statement.

So really they don't need to see the card, they need to know that it was your card when you booked. So a photo of the card or a paper statement would probably do as well.
I think I got flagged because it wasn't my normal travel pattern. Of course the only time I have ever been asked (in 25 years of semi-regular flying) was the one time I didnt have the card
Why a passport with the same name on it isn't good enough I don't really know.