That comment has been an itch in my brain for a long time.
Obviously there is the explanation of: "Well, they filmed it in the 1960s, and they wanted to use terms and phrases which could connect to the audience, so obviously some technology will look outdated!"
That explanation is fine and good. It's why many sci fi writers stick to conceptual presentations of scientific theories, rather than details of how X item works. You don't have a crystal ball, and any detailed explanation will ultimately date you and your writing. This is a delicate balance of maintaining the suspension of disbelief, while staying connected to the audience that must be achieved.
However, there is something else at play, a factor which writers and producers can use to their advantage beyond "maintaining the suspension of disbelief, while staying connected to the audience".
There can be a real fictional world explanation. That explanation is found in the history of Star Trek.
In the 1990s the world governments were overthrown with despotic eugenics-born tyrants. These men and women threatened all of humanity with their dictatorial rule. It was during this period where the world gained limited space travel, using technology very similar to that of the 23rd century. Normal Citizens fought a bloody uprising, throwing these leaders out of power, and the power vacuum involved brought forth World War 3, destroying most of civilization.... this in turn brought about the post-atomic horrors, until that was broken by even more unrest and warfare. This ended with the first warp flight into space, and First Contact with Vulcans.
It is a simple enough job to take this history of Star Trek and apply it to the available technology of the 23rd century. Humanity has just lost resources, time, manufacturing and an entire civilization to 200 years of prolonged war. It makes sense to use that as an explanation for why advanced resources were used for the more critical and fundamental aspects needed to survive and progress, while older technology is utilized for the non-essentials.
At the same time, the "Tapes" that are used in Star Trek the Original Series, are clearly far more advanced than VHS. Hours of time indexed video, with full ranged audio, are maintained on what "appears" to be a simple floppy disk. We didn't have that technology in the 1990s, although we do today.
The point of all this is really an exercise in writing and story telling.
As a writer or author, you don't always NEED to come up with an explanation, if there's enough content in your piece to explain how the world is and how it operates. Don't make your characters give exposition for everything. They LIVE in your world, they do not need to be told HOW your world works. Balance that out with instances where it's natural for characters to expound on a thought while referencing their own history, then let your readers learn through their own path of discovery and debate. At the same time, it never hurts to create an encyclopedia of your work, for yourself AND your readers. Make your personal encyclopedia (or bible) more indepth, and give your readers something that's lighter to consume.
At the end of the day, writing is about discovery, exploration, explanation and a guided framework. Your readers need to use the frame you construct to add on and explore. Science Fiction serves that purpose, not just for readers, but also for goals that you believe society should strive to achieve. Science Fiction is at its best when exposition isn't bald, or trite. It's at its best when it inspires and encourages growth into discovery.
So next time you hear a sci fi show talk about tapes, or use a console that's reminiscent of Mercury Space Program, don't immediately blow it off as just being an artifact of the time, stop and think if there might be something there which makes it a necessity in that fictional world. It just might give you more insight into the work than you had before.