How does Gyro Gearloose's security system work?

Solution 1:

According to that wikipedia page,

The Amiga and DOS game features copy protection in the form of a unique password system. When the game is booted up, you must use a key script (located in the game manual) to translate symbols on the screen. [..] there is no way to play the game without deciphering the code.

A copy of the manual can be found here.

However, just a side-note: it is possible to break the code without the manual.

Note that in your 3rd and 4th screenshots, both codewords begin with a 'C'. Given the choices available in each of these, this means the code-word 'C' must be deciphered to either F, R, or W.

Additionally, pictures 2 and 4 contain an '8'. Given the choices, '8' must be either M, G, U, or V.

Combining the previous two paragraphs and looking at picture 4, we know either 'C'=R, '8'=U; or 'C'=W, '8'=G.

If you continue this logic using more screenshots of codewords with matching characters, you'll eventually decipher every letter.