My level BM 50 hunter has leveled up with both a boar and cat for the last 20 levels (alternating, obviously, since you can't have 2 pets out at once).
The cat is considered by some as more offensive and the boar as more of a defensive pet. I have not tried a birdy or wolf except to learn abilities for future pet taming.
I have noticed little difference in leveling/grinding speed between the cat and boar or how many mobs they can take on at once. Their differences, for me, seem to balance out.
The cat (claw, bite, dash, growl) does drop mobs a tad faster, but it requires a little more mana sometimes for extra heals compared to the boar. The boar (gore, bite, charge, growl) does good damage, too, and its charge is on a shorter cooldown than the cat's dash.
I generally don't try to take on more than 2-3 equal level mobs at a time with either pet. Just more micromanagement than I like at one time. Four mobs are doable without much sweat with either pet, and probably any pet, if I cycle through the mobs to get aggro on each so that healing aggro from pet mend wont draw mobs away from the pet. Don't remember trying or having to drop five mobs at once.
A bird with AOE screech might be able to hold healing aggro without having to cycle through and attack every mob before casting pet mend. I just don't have any experience in how much healing you might be able to do before exceeding the threat generated by screech. Obviously, feign death will help, but the more mobs you fight the greater the chance of a resist, which I personally hate when I am just on auto pilot while leveling.
Another consideration is what pet you like. Maybe you don't or won't have a personal preference, but when I've grouped with hunters with birds on my other toons, sometimes the flapping wings got on my nerves.
(I am the type that is about to ditch [for leveling, might use it grouping] the level 48 epic hurricane bow for a 51 blue crossbow because the hurricane fires so annoyingly, to me, fast. The hurricane fires almost once a second with talents. It's purple, but I am not too attached to it because I got it for just 50g. I also never pick a race for a toon that I can't stand to look at for 70 levels plus end game stuff.)
I may get a bird when I hit 70 (the unique carrion bird in terrokar seems tempting [trachela?]), but I don't wanna level with one regardless what it can do just because I really like my cat (ghost saber) and boar (rotting agam'ar) and they get the job done for me.
Trying different pets and seeing which one works for your play style is part of what makes a hunter cool for me. Enjoy.