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.
Good examples, I'll use them!
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.
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.
Actually, we did discuss magnetic tapes and the analogy to the microtapes. And, we talked about floppy discs, but didn't think of the audio tapes.
However, you're incorrect, punch cards were used up to the 80s believe it or not. During the 60s, magnetic tape was starting to take off but punch cards were still widely used.
And, floppy discs were just about to go on the market. First floppy discs were introduced in 1972 and obviously were being developed around the time TOS was on the air.
However, that's all kind of besides the point, at least for our debate. We were debating functionally what were the microdiscs capable of and what were they more similar to.
Thanks for the examples and audio cassette analogy, very helpful!