I think that hunter pets like mounts should be account wide. I love playing hunter hence why i have 3 of them, and it really sucks having to go around to find and tame pets again and again. Especially if you like to have rare pets as well. On my one hunter, i’ve put in some ridiculous amounts of time waiting for a rare to show up so that i can tame it. If your thinking i’m putting in that amount of time again for each of my characters, that’s all i’d ever being doing then. There’s no reason why and can’t be as well. There’s no reason why pets can’t be account wide just like mounts are.
It would be cool if the stables were like a bank that all your hunter alts could use. However if you put pet A in and take it out on Hunter 2, then you should have to put it back into the stables to take it out on Hunter 1.
They didn’t do that with mounts. Any toon can call any mount from his stables, with as many toons as he/she wants.! Why do it for combat pets?
Only an issue for people who keep repeatedly leveling hunters, which is a more pressing problem to address.
Your problem is pretty niche.
What you have to realize is some of the hunter pets are level specific. They aren’t going to let a level 15 toon run with a level 35 pet. I understand how it would be nice and EASY to share the same pet stable but then you really wouldn’t be a “hunter”. Good luck! 
As much as I like the idea… I still prefer to have pets specific to that character. My dwarf hunter uses the slime that was retired back in wrath. Also has the yellow firefly from Zangarmarsh. My troll hunter uses raptors/dinos but refuses to use a devilsaur because… reasons. I have a Lightforged Dreani that uses stags. Made a new goblin that is going engineering and will be using mechanical pets… Not many of them low level, and the ones available at low level are chickens and rabbits. There is that one sheep… but you have to be able to fly to get to him. Looking up those pets… most are found in alliance held areas… so will have to dodge guards.
It would be nice to go out tame the pet and be able to use it on another hunter… but part of the class is the thrill of finding that one cool pet that you can have. (cool to you that is)
Mounts go into a collection, not a stable. Pets also affect combat while mounts do not. It’s not an equal comparison.
I think some of the hunter pets should double as mounts .
The site of a hunter riding into combat on a bear or wolf and both working as a team to combat their foes.
Would they increase the limit even more? Across all of my hunters, I definitely have more than 350.