The animals in Pillars of Eternity 2 Deadfire can be divided into two categories: animal companions and pets. The animal companion is available only to one character class: the Ranger. A ranger can have one animal companion (you can choose a deer, bear, wolf, lion, boar, antelope), which comes with its own skills and stats - in addition, you can control them like any other party member - so having an animal companion increases the party size. The animal companion is chosen during character creation (the choice is permanent).
On the other hand pets can be found during world exploration. They are spread all around the game, so if you don't want to miss them, you have to be very observant (it's helpful to use the "Tab" key that displays the names of nearby items, NPC, pets etc.) Pets don't take part in combat, however they offer a stat bonus (the bonuses vary depending on the pet type). If you find a pet, you "put" it in your inventory like any other item; if you want to use it you have to place it in a particular slot (just like weapons and armor). The pet will appear at your feet and follow the team, it is immortal but doesn't deal any damage. You can "use" only one pet at a time, but all the others you have found may safely reside in your inventory, in addition, a pet can be owned only by your main character.
You can find the first pet on the starting island (Port Maje Island), specifically at the Engwithan Digsite, you will visit this location while doing the Stranded and To Hunt a God quests. The cat, Gosha is located near the main entrance to the Ancient training room (see the map above). Gosha grants +1 perception to your main character and reduces the chance of getting critically hit by enemy weapon attacks.
The next three pets can be easily found in Neketaka, namely in Queen's Berth (M11). See the three entrances to the buildings marked on the map above:
-  Vailian Trading Company Headquarters - head to the terrace (the entrance is located on the first floor). There, you can find a pet called Toby - it gives you +1 Perception to your main character and a health bonus to the entire party.
-  District Home - Xopn'aua can be found in the small room inside of the home. This pet grants you +2 Insight and +2 Metaphysics to the main character as well as a bonus to Will for the entire party.
-  Luminous Adra Mill - pet called Abraham can be found on the huge balcony. This animal reduces Armor Recovery Time and allows your party to get a little bit of health back when they kill an enemy.
The list of other pets can be found on List of pets page.