We do know for a fact that data can be recorded on the tape cards; for instance, in "Wink of an Eye," Kirk records his message to Spock on a microtape and leaves it where Spock will find it. There's also Kirk's "last message" tape from "The Tholian Web." Earlier, in "Balance of Terror," we saw Uhura giving Spock a microtape containing the recording she'd just made of the transmission from the Romulan ship.
Remember that at the time, computer data was stored on magnetic tape, which was why TOS so often referenced "data tapes" and the like. But that tape was stored on large reels, like on those old-timey computers you used to see in TV and movies. The microtapes were presumably meant to be a more compact form of computer tape, in the same way that audio cassette tapes, which were a brand new, cutting-edge technology at the time of TOS, were a more compact version of reel-to-reel audiotape. Basically the makers of TOS were combining the ideas of computer tape, audiotape, and videotape (which did exist at the time and was used in the TV industry) with the "futuristic" compactness of the audiocassette and projecting it forward to an even more compact tape cartridge.
So both you and your friend are focusing on the wrong portions of the history of computer storage; punch cards were too early and floppy discs too late. They were extrapolating from the tape storage media that were standard at the time.
(I'm reminded of the origin of the tricorder -- it was an extrapolation from the portable audiocassette recorders that were cutting-edge at the time, but with sensor and computer functions added on, hence a tri-function recorder. Audiocassettes are so quaint to us that we don't realize how futuristic they seemed when TOS was being made.)