Back then, tapes were just binary data on a magnetic film, with no "hidden" channels or out-of-band capabilities. Manufacturers that claimed to make tape-to-tape recording impossible often just made the tape look different, to deter would-be pirates. A regular tape recorder module was usually used to read them, so making "special" tapes couldn't really work.
The usual method was to ship a booklet (e.g. an instruction manual) with multiple pages, then ask the user to input a particular word from a particular line on a particular page. Code cards were also used, utilising embossed black text on dark paper to prevent easy photocopying.
Anyone could copy the tapes, but copying the booklet was more difficult at that time; nobody had scanners, and few people would go to the effort of photocopying a multi-page booklet at their local library. Even if they did try to copy the booklet, key words would often be darkened in the text to prevent them from being copied cleanly.
Some early video games included merchandise, code wheels, text on CD cases (e.g. Meryl's codec number in Metal Gear Solid), or other physical goods that were required to play the game (e.g. to solve certain puzzles), and these were reasonably effective deterrents. Other games simply shipped posters and action figures to make the original copies more valuable than a copy could be.
As a (slightly off-topic) side note, VHS tapes did have the ability to contain data that couldn't be copied by normal players. Essentially a VHS player works by running the digital signals from the tape into a digital-to-analog converter (DAC), which then sends that signal to the monitor along with an appropriate synchronisation signal. This analog signal is interpreted as a set of colours, but some subtle peaks were also used to send commands to the player (e.g. blank screen, reset vertical sync, etc.).
Macrovision (a copy-protect mechanism) encoded high voltages into the tapes, such that the player's protection circuitry would compensate. On the original tapes, the value would be interpreted as a glitch and be ignored by the player. On a copy, the recorder would reduce the signal level to that of the rest of the video. Once that part of the signal was reduced to a normal level, the player would interpret it as a blanking signal, causing the video to be entirely blank (usually blue). Other mechanisms relied on the same trick, but instead used signals that caused the video to be hue-shifted, or that caused most players' tracking mechanisms to malfunction.