Why pairing codes are single-use
A pairing code is single-use and short-lived by design, generated fresh each time you open a Remote Control chat in the web app specifically to pair a new machine.
It is not meant to be a reusable credential you would save somewhere for later; once a machine has successfully used a code to pair, that code has served its one purpose.
Pairing a second, different machine requires generating a new code from a fresh Remote Control chat, rather than reusing an old one, which keeps the pairing mechanism from becoming a standing secret worth protecting long-term.
This single-use design limits the damage of a code being seen by someone else in passing, over-the-shoulder, a screenshot, a shared screen during a call. Even if that happened, the code is worthless the moment it is either used once or a fresh one is generated in its place.
There is no way to request a pairing code without going through the Remote Control chat in the web app; it cannot be generated from the tCode terminal itself, which keeps the machine being paired from being the one deciding whether it gets to be paired.