There's always the possibility that they're some other kind of storage medium but everyone just calls them "tapes" out of old habit, much as we may still speak of "taping" a show off TV, or for TV, when it's not really tape being used.
Well, yes, of course. I don't think anybody here is claiming that we believe there "actually" were tapes inside the cartridges. We're not pretending the Trek universe is real and trying to rationalize it; we're talking about the thinking of the people who lived in the 1960s and created a work of fiction called Star Trek
. The point is that they
assumed they would be actual tapes, because at the time they were writing these scripts, magnetic tape was the vanguard of computer technology.
You see the same shortsightedness about progress in a lot of science fiction. Read SF novels and stories from the '40s or '50s and you'll see the writers assuming that people thousands of years in the future will still use punch cards or wire recorders or microfilm, or that computers would always be vast, room-sized agglomerations of vacuum tubes. Aside from Murray Leinster's "A Logic Named Joe," which kind of predicted the Internet, most SF writers assumed that an entire country or planet would have a single, gigantic central computer that would act as an oracle granting answers to questions put into it, usually in punch card or magnetic tape form. They were generally as bad at predicting the advance of computer hardware as they were at predicting the advance of gender equality.
Although the problem isn't limited to that era. TNG predicted the tablet computer (the padd) and the flash drive (the isolinear chip), but it didn't predict wireless networking; people still carried padds or chips around physically to deliver data to each other. Nor did they predict that the functions of communicator, padd, and tricorder would end up being combined into a single device.