Yep, pet’s can’t have pets or buffs. The way the game works would make some unfortunate situations.
IE - a pet with a pet buff, does not affect itself, but it’s own pets.
Or, a pet with pets - that pet isn’t yours so your buffs do not affect it, and anything it kills doesn’t give XP to you, but to your pet (not that it would do anything useful), and who knows what other weird quirks might happen.