A very interesting idea for a thread! Takes me back and makes me feel all nostalgic haha. I'll give this a go.
Expansions
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Vanilla: Cats ruled Vanilla WoW. Though there were exceptions (Bjarn the bear, Aga'mar boars, and Takk the mount-killing raptor), the pets I think back on as being most representative of the times were Humar the Pridelord (my main still has him and always will), Ghostsaber, Rak'Shiri, King Bangalash (Cobra Reflexes anyone?) and Brokentooth, the original game-breaker and bane of mages everywhere.
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BC: This expansion was a disappointment for me in terms of pets. I felt like there were alot of missed opportunities in terms of pet families that could have been implemented (basilisks, clefthoofs, etc.), and families like Sporebats and Nether Rays that despite a small dedicated following are considered largely irrelevant but most of the community. But it did give us the impossibly ugly ravagers, which for a brief time eclipsed even the mighty cat family in terms of sheer dps. In terms of iconic pets of the Burning Crusade, I would say ravagers, warp stalkers, Ironjaw the wolf, and Hawkbane and his Blackwind Sabercat minions.
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Wrath: Wrath opened alot of doors for our chosen class. A bunch of new families were suddenly tameable, and in particular exotics shook things up for how hunters could influence raids, pvp, etc. I don't think Wrath can be discussed without Spirit Beasts, Devilsaurs, and Corehounds holding a position of honor. The frenzy that surrounded Loque'nahak in the early days was without equal until the sad displays of greed and vindictive play that heralded the arrival of the 4.2 challenge tames. In Wrath, the Furious Four (Loque, Skoll, Gondria, and Arcturis) remain in demand to this day, and the Sholazar Trio (King Krush, Aotona, and Loque once again) still drag people back for maddening flights around the Basin. Also, the Shattered Hand Warhound gave people hope that we might be able to tame other armored mount-style pets, but to no avail. But the issue that I feel like is still a sore subject for many hunters like myself was the removal of our Garwal worgen tames. Yes, I understand the logic behind it, and what's done is done, but he was mine - tamed fair and square - and I loved him, and emotion does not respond well to logic

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Cata: Cataclysm shook things up again. I feel like our pleas have been heard in alot of ways, and the developers went above and beyond in giving us what we asked for. Many skins previously not in the game - notably cats, chimeras, ravagers, and wasps - were added to replace rares with common models or just mobs with repetetive skins. Also, a lot of new pets had vastly improved graphics - Sambas next to Humar or Echeyakee is particularly painful - and a variety of other families gave us alot to play with and enjoy. I would say Sambas, Karoma, Madexx, and Terrorpene were the Holy Grail of early Cataclysm. Since then the 4.2 rares have joined the ranks as symbolic of the expansion's contributions to the game.
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Alliance Races
Alliance in general: Sambas paired with the Golden King Alliance Guild Mount is poetry in motion.
Nelf: Saber cats are linked forever by lore (and now starter pets) with this race. This is unfortunate for me, as I hate playing Night Elves, but I digress. Beside the cats, I associate birds - particularly eagles and owls, as well as the druid model "hawk" - with the Night Elves. Bears and foxes also get a mention.
Human: Humans can be paired with either the conventional, "real-world" hunting companions - mastiffs, wolves, and eagles - or the plain look of a human can be used to emphasize the appearance of the more absurd, crazy pets that the game offers. I personally think humans look good with raptors, rhinos, saber cats, spiders, turtles, warp stalkers, and wasps.
Dwarf: The Dwarves are creatures of the cold, mountainous regions of Azeroth. They also have ties with stone and earth. The most iconic pets for them, at least early in the game, are frostsabers, polar bears, and snow leopards (though I think Duskstalker's saber model snow leopard looks best), but I think any shaggy pet does well and matches the heavily-bearded dwarves well. I would include chimeras (old-world), gorillas, lions (particularly Sambas and his massive mane), and rhinos.
Draenei: Draenei have an alien look, and I think the most iconic pets for them come from Outland. Chimeras, Core Hounds, Dragonhawks, and Ravagers all apply.
Worgen: Worgen are a fun one for me. Obviously mastiffs fit well with the whole "Werewolf in London meets Jack the Ripper" vibe that Gilneas has, and wolves, particularly worgs and saber worgs, need no introduction. However, I personally think worgen faces are more leonine than canine, so lions and saber-cats look great with a Worgen hunter as well. There's just something about a werewolf and a sabertooth tiger running side-by-side that just feels right

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Horde Races
Horde in general: Any of the many Madexxes skittering along beside a Kor'kron Destroyer is wicked.
Orc: I've rambled a fair bit, so I'll try and keep this brief. Boars, devilsaurs, tigers, ravagers, Northrend worgs and worms - basically anything with tusks

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Tauren: Tallstriders are most people's iconic tauren pet. I would argue though that as the most imposing hunters physically, the hulking Tauren need a big pet they won't break accidentally. They are also in touch with nature. Big natural beasts (ie not core hounds or silithids) like devilsaurs, rhinos, spirit beasts, and worms look like what a tauren would want beside them. If we're talking non-exotic, beetles, gorillas, saber cats, and wasps all are big and strong-looking enough to be tauren's rough and tumble hunting partner.
Undead: Creepy pets and undead go hand in hand. Demon dogs, the plagued bears and wolves, spiders, and the undead boar get special mention.
Belf: All I can really say beside the obvious dragonhawks and lynxes are pets that match Blood Elves in one of their two most distinctive traits, beauty (cats, parrots, spirit beasts) or cruelty (core hounds, spiders, ravagers). Just my opinion though.
Goblin: The same rationale as Tauren, but for the opposite reason. Being so small, goblins need an enforcer. Huge pets, like core hounds, devilsaurs, rhinos, silithids, and worms look like what a goblin hunter would make a beeline for to tame. If we're talking non-exotic, beetles, gorillas, saber cats, and wasps all are big enough to be a goblin's muscle.
Troll: Trolls ARE raptors and tigers. Rinse and repeat. Lol no, I also think crocolisks, devilsaurs (giant raptors!), dragonhawks, gorillas, lynxes, and serpents all look right next to a troll.
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General
Epitome of a "newbie" hunter: I would say whatever pet each race starts out with now, as I've actually come across hunters now who didn't know you could tame other beasts... sigh.
Epitome of an "experienced" hunter: In a raid, whatever pet covers the missing buffs. Another, less effective scale of rating experience is someone with old-school rares at 85 (tells me they've been playing with that pet since Vanilla and are seasoned veterans), those that have the harder to get challenge tames (Ban'thalos, Chromaggus, Deth'tilac, etc.).
BM hunter pet: Spirit beasts, devilsaurs, and core hounds, in that order.
Non-BM hunter pet: Generally I see cats, dragonhawks, spiders, windserpents, and wolves. Extra emphasis seems to be placed on rares across the board.
Show-off pet: Obviously rare spawns are the definition of this, but as I said before certain pets to me have a "wow" factor due to looks, reputation, or difficulty of tame. Off the top of my head: Deth'tilac, Ghost Hydra/Ooze, Grimtotem Spirit Guide, Sambas, Chromaggus, Loque'nahak, King Krush, Oiled Worg, Garwal in Worgen-form (before he was hotfixed), Terrorpene, and a number of others that are slipping my mind at the moment.
If you made it all the way through my rambling, I applaud you and appreciate it!