Mechanics: Monster Catching

Pokemon is far and away from being
the only or oldest game to implement
catch 'em all mechanics.

I'm going to riff off of a topic that came up on a forum I participate in. The question, specifically was for a deterministic translation/formula to convert pokemon stats into GURPS, which, I suggest is unfeasible for a number of reasons, and probably will give you neither a fun GURPS or pokemon experience. Instead, the better idea is to try to rework the mechanics of that or similar games in a way that makes sense in the GURPS framework.
That said, being a trainer of powerful creatures, monsters, demons, etc. isn't a completely unique mechanic to pokemon, and in a very light way, it is touched upon in GURPS here and there, so we aren't starting from nothing. Let's see what we can scrounge together to implement a system for corralling powerful allies.