yes, you need to show the card used while booking to get the physical tickets.
On ticketplease.com they have mentioned that they accept a photocopy of the front and back of the card in case the cardholder is unable to attend the show. confirm this with the movie hall to be sure.
And the buy one get one free is applicable for any Signature Visa cards, not just ICICI. It is also restricted to just one free ticket per month.