The Pets buff the already existing Team Bonuses, rather than providing new ones. I wrote up a brief explanation here: "Magic Bonus" is either broken or worded poorly - #8 by Ozball
Also @Rickygervais I’ll need to double check, but I believe it should save your Pet Rescue team across the same colour Pet rescues (eg all Purple {Type} Pets will have the same team saved when you enter them)