Are there any significant animal characters in WoW aside from Rexxar’s bear Misha and his other animal companions?
My mind first goes to the Frostwolf Clan, but I haven’t memorized any of the wolves’ names.
(I am not referring to Wild Gods or Celestials)
Would Al’ar count? “Phoenix God” is a lofty title and all but it’s best remembered as “Kael’thas’s beloved pet phoenix.”
Most of these aren’t really significant so much as they are reoccurring.
Among the Alliance:
• Anduin has a mount named Reverence, plus a newfound Fox friend as of the recent short story.
• Tyrande has Ash’alah, which is her nightsaber.
Additionally, she has two owls, the first of which is Dori’thur who is keeping an eye on Sylvanas. The second is Andi’thur.
• Falstad Wildhammer has his gryphon, Swiftwing.
• Taelia Fordragon was never far from her gryphon, Galeheart.
Among the Horde:
• Thrall has his mount/companion, Snowsong.
• Talanji has a pet raptor by the name Tze’na.
• Lasan Skyhorn often rides on his eagle mount, Aviash.
Neutral:
• Lorewalker Cho has a pet cloud serpent called Mishi.
• Shen-zin Su, the Wandering Isle.
• Ozumat
Rexxar has Misha.
And the PC can tame the Loa Tiger in the Emerald Dream.
I can’t believe I forgot about the PC’s Dog.
There’s also Hati, who helped BM hunters fight the legion during Legion and later joined us permanently as a companion in BfA
Can’t believe everyone forgot about the best boi, Hati. So sad 
I mean, lorewise if everyone has them – They kind of lose their significance as a stand-alone character. 
To some extent I suppose you could do the same with Hati.
As for the reply towards the OP / thread — Jintra just about covered it. 
I love Jintra’s post. He did an awesome job covering most of the big name famous pets.
That being said, Hati being Thorims companion at one point, I just thought he was worth mentioning since Thorim is a pretty well known character to most players 
Emmigosa was given to you as a battle pet in Legion, but she showed up independently in DF. Maybe she canonically stayed by your side for a bit then left.
I’d love if they applied some degree of dialog towards that.
“Emmigosa is a tricky one … But also highly intelligent for their age. It appears in their travels alongside you, she forged a simulacrum to take her place and parted ways.”
That way you could still see them out in the world, yet also keep lore-significance towards them via the pet.