While you can only get a select few permanent partners, there are some Pokémon that, when you are battling them, will ask you if they can join

If you say yes then you will get to pick it's name
Once you've picked the name this Pokémon will follow you only in this dungeon. However if it faints, you wont fail the mission like you would if you or your original partner do
This can randomly happen in any dungeon with any Pokémon so be prepared while battling!
If they survive, you can go visit them and check on their stats in various wild places around the island which you can get to by going West in town! Click here for a list