I'm building a prototype for a game that would have a lot of different companions/pets. The idea is to be able to swap out the active pet for another from the player's party.

I'm having a difficult time figuring out where to begin on creating the party for the player to choose from. Would it be similar to having different weapon slots?

Any help is much appreciated!

Depends on the big picture. Conceptually it's a tag on characters, and a party is composed of things with the tag.

That could be implemented as a type of slotting system, if that is what you want. You could design a weapon system with tagged weapon data in a similar system.

Implementation-wise, you're going to have the data somewhere. You'll probably have a small structure in the GameState or GameMode about the parties that links to conceptual objects that get chosen between. There are lots of potential ways to build it up, with details depending on your goals overall.