Ignoring how powerful hunter pets are, this is actually a really bizarre issue that’s been plaguing classic for apparently years without a fix:
It can also cause weird issues when you reconnect that look odd to other players:
I think I noticed this once when playing around with on my hunter (5th 25). After reconnecting I jumped off a tower I was on and after running a ways everyone vanished and I thought I was lagging. I re-cast eyes of the beast since my pet wasn’t following me and surprisingly, my pet was still in the tower… next to me! I cancelled it and was back in the tower.
Eyes of the Beast also bugs out the pet bar so you can’t see which skills are set to auto-cast after using it for some time or until you reload your UI.