Just to clarify a few things here:
The color of the pet doesn’t directly have any connection to what it buffs, it specifies what kind of food it likes to eat.
The Bonus card tells you which team bonus it buffs (the bonuses you get for using unique troops of that type/kingdom).
Example:
Feyrie is a Purple Buff Pet. Therefor she likes eating Purple Food (Purple food will have a star next to it in the Level up tab and will award more XP to the pet). She is also a Buff Pet, meaning she buffs one of your existing team bonuses. (Loot pets buff your Souls/Gold/XP, Event pets are just Cosmetic)
Looking at Feyrie’s Bonus Card we can see she buffs the Fey Team Bonus, which you get for having 3-4 Unique Fey troops on your team.
Looking at this example team I put together, you can see we have 3 Team Bonuses active for it:
Lord of Summer - 2 Unique Bright Forest Troops
Lord of Thorns - 2 Unique Forest of Thorns Troops
Fey General - 4 Unique Fey Troops
You can see Feyrie displayed in the Fey General section to show that that bonus is being buffed by her. (it is normally +2 Attack and +2 Magic) If I had Twigant as well you would see him showing up on the Lord of Thorns section and an increase to those buffs.
TL;DR: Pets buff the existing team bonuses mentioned on their Bonus card. The colour of the pet isn’t directly tied to what they buff. (Though you will usually see like with like, eg a Purple team bonus pet is generally purple etc)