In Monster Hunter World you can unlock pets for your own room. Your room is found in Astera. It’s one of the more hidden game mechanics.
To unlock new pets you must capture small animals with your capture net. Don’t confuse them with small monsters. Potential pets are tiny creatures that won’t attack you like monsters do. An example is a hare. You’ll see it and many other small animals jumping around the open world quite frequently. So head out on an expedition, select the capture net from your inventory, press / to aim and / to release.
Then back in your room, talk to the housekeeper to release the pet!
The capture net is automatically placed in your inventory at the start of the game and can be used infinitely.