Expansions
Vanilla: Humar, definitely. He was *the* go-to 'awesome pet' back then.
BC: Any cat. With them being the 'big damage pet', they were everywhere. 'Cat or GTFO', pretty much, was the idea. Raptors and windserpents got their turn as well, but cats were the big ones. ... I raided with a windserpent, myself.
Wrath: Wolves. Wolves, wolves, wolves, wolves. Everyone had a wolf. Beast Masters, Devilsaur. Loque was definitely the 'big name', though.
Cata: Sambas and Madexx. A lot of people like Terrorpene, Karoma, et cetera, but those were the iconic ones. Later in Firelands, the 'rare spiders', of course.
Alliance Races
Alliance in general: I can't think of a general one here.
Nelf: Owls. I personally think they should have started with an owl rather than the white sabrecat (not that there's anything wrong with white sabrecats). When I see night elves, I always think 'owl', though.
Human: Dogs. Wolves match well, too.
Dwarf: Bears, boars, and snow leopards.
Draenei: Almost any of the Outland pets. Ravagers, in particular, since you can get them at low level. Moths match well, too.
Worgen: Wolves. I don't care if they start with dogs. They should start with wolves!
(Not including Gnomes as they're not a hunter class, but feel free to add something if you wish): Gnomes, for me, would work with mechanical things better than hunter-pets.

If you made me choose a hunter pet that matched them, though, I'd say give Draenei the ravager and give gnomes the moth.
Horde Races
Horde in general: Don't really have a particular one here.
Orc: Wolves (especially worg-style), boars, and scorpids.
Tauren: Tallstriders. They look amazing with an owl as well, but they're not exactly easy to get.

Undead: Spiders, and of course any of the 'dead/dying/rotting' pets.
Belf: Dragonhawks, lynxes, brightly-colored birds. Almost any pet that is really showy.
Goblin: I'm not sure of what I would consider 'iconic' for these.

Troll: Definitely raptors. To me, they couldn't have started with a better 'newbie' pet. They also go well with jungle cats. And, while I know the lore doesn't allow it, they look amazing with birds.
General
Epitome of a "newbie" hunter: Any of the pets that are from the area they are in. Also, boars. boars, back in vanilla/BC days were the iconic leveling pet because of how well they would hold aggro ('Charge' would double the aggro of your opening Growl).
Epitome of an "experienced" hunter: Any hunter who carries the pet with the 'buff needed' in a party/raid. I don't really consider taming one of the 'Firelands' rares to be the epitome of an 'experienced hunter' since there's a ton of tactics online and people can go to their friends to help out for the most part. I won't deny they take *skill*, but I wouldn't call them *experienced*, if that makes sense!
BM hunter pet: Devilsaur and Spirit Beast.
Non-BM hunter pet: Wolf and cat.
Show-off pet: Any of the hard-to-get rares or FOTM 'special skinned' rare/pets. Also, any 'special tames' that have been recently discovered (Garwol, fire pets, buff-pets, impossible skins, et cetera). I can't think of a particular one here because it changes so often.