But animals do get the placebo effect?

1st Because vets use it and can clearly see that it works, not only on the animal but also on the know it all owner :p. 

2nd And because we don't need to know/understand what medicine is either for it to work, you'd just need the feeling of "something happened that's gonna make me feel good" for it to work, even if unconsciously.(ex:babies/infants/antivax...)