Spirit Animal is a party perk (picked once, affects all animals from all companions).
Strategy[]
Leveling up the Animal Whisperer skill and taking the Animal Training perk will increase the constitution of a character's pets, helping to avoid their deaths.
If you have pets or followers, avoid using area of effect attacks that apply damage over time, like molotov cocktails, because they don't avoid the area of these attacks and might die because of it.
Every time you travel to another map or reload the game, your pets and followers will be at full health, that is the easiest and fastest way to heal them.
It's possible to have more than only one pet per character, since some pets are recruited through dialogue; however if any character that has more than one pet is removed from the party, the pets won't be tamed anymore and you won't be able to tame all the pets again with only one character. To maximize the size of your pet legion, first tame any pet that isn't in the list bellow (I recommend Billy and Jean, because technically they are two pets in one), then:
Major Tomcat: tame it through dialog (if it is not in the map so you can talk to it, just take it from the kennel and dismiss it first, so it stays in the map);