Just always interested in hearing different thoughts! Particularly if you can manage minimal overlap (Demon Hunter pretty much making it impossible to fully avoid that, though).
Also, this is NOT Spec-specific. Rather, taking the class “as a whole”. A Priest, for example, needs to be a Race that’s “fitting”, whether it be Holy, Discipline, or Shadow.
What would you pick, and why? Which do you consider the best picks for each Class, or which the most trouble to decide?
Every single class to one race or… one from both Horde and Alliance?
I think there will be a lot of overlap in general, given the sheer number of playable races.
For hunter, I pick dwarf with bear pet.
For Druid, I pick night elf.
For Shaman, I pick that horde one with the funky ear things.
For paladin, I pick human.
For mage, I pick gnome.
For monk, I pick Pandas.
For Rogue, I pick undead.
For Warlock, Priest, and Death Knight, I pick whoever isn’t mechagnome or kul tiran.
I forgot another class but if I forgot it, it’s probably not important so 
.
Horde:
Orc: Shaman
Tauren: Warrior
Troll: Hunter
Undead: (Warlock)/Deathknight
Goblin: Rogue
BElf: Mage
ZTroll: Paladin
HighTauren: Hunter
Nightb: Mage
Vulpera: Hunter
Alliance:
Human: Paladin/Deathknight
Dwarf: hunter
NElf: Druid
Gnome: Mage
Draenai: Shaman/Priest
Worgen: Warrior
Velf: Idk
LBDraenai: IDK probably Paladin
DIDwarf: Warrior
Kul Tiran: Shaman
MGnome: Hunter (with mech beast), or maybe warrior
Neutral:
Panda: Monk
Dracthyr: Evoker
Warrior: Orc
Paladin: Human
Hunter: Night Elf
Rogue: Goblin
Priest: Human/Troll
Shaman: Tauren/Orc
Mage: the Highborne (i.e. Blood/High Elf, Nightborne, and remaining Night Elves) and Draenei
Warlock: Orc
Monk: Pandaren
Druid: Night Elf (However, considering the infusion of more solar/sun imagery Tauren?)
Death Knight: Human/Forsaken (they’re both undead!)
Human Paladin, Dwarf Warrior, Gnome Warlock, Night Elf Male Druid Female Priest, Draenei Paladin, Void Elf Shadow Priest, Nightborne Arcane Mage, Highmountain Tauren Elemental Shaman, mulgore Tauren enhancement shaman, or c warlock, mag’har orc shaman, etc etc
Warrior: Tauren
Paladin: Tauren
Mage: Tauren
Rogue: Tauren
Demon Hunter: Tauren
Druid: Tauren
Warlock: Tauren
Death Knight: Tauren
Priest: Tauren
Hunter: Tauren
Monk: Tauren
Shaman: Tauren
Evoker: Tauren
undead = warlock
human = pally/deathknight
gnome = rogue
dwarf = warrior/hunter
orc = warrior/shaman.
blef = mage/warlock
nelf = druid/priest
drenei = shaman/pally
troll = shaman/hunter
tauren = warrior/druid
pandaren = monk
worgen = hunter
goblin = rogue.
Warrior: Tauren. They’re huge. Orc is a close second due to warrior culture.
Paladin: Draenei and human tie.
Mage: Blood elf.
Rogue: Night elf. They’re literally the night race. Nocturnal, have Shadowmeld, harder to detect through stealth, close to nature so good with poison.
Demon Hunter: Night elf. They invented it. Still the best.
Druid: Night elf. Original druids, best druid lore and aesthetic.
Warlock: Forsaken and orc tie.
Death Knight: Human. Because Arthas, and it was originally just humans.
Priest: Draenei and human tie.
Hunter: Tauren and night elf tie. Bows are big in night elf culture, hunting is big in tauren culture. Both have strong bonds with nature and beasts.
Monk: Pandaren.
Shaman: Orc and tauren tie.
Evoker: Dracthyr. This was a hard one, but I think I made the right choice.
Oh boy. Get ready for these hot takes.
Alliance
Human – Mage
Night Elf – Druid
Dwarf – Hunter
Gnome – Warlock
Draenei – Paladin
Worgen – Warrior
Pandaren – Monk
Void elf – Mage
Lightforged – Paladin
Dark Iron – Shaman
Kul Tiran – Rogue
Mechagnome – Mage
Horde
Orc – Warrior
Undead – Rogue
Tauren – Druid
Troll – Shaman
Blood Elf – Warlock
Goblin – Rogue
Pandaren – Monk
Nightborne – Mage
HMT – Hunter
Mag’har Orc – Shaman
Zandalari – Priest
Vulpera – Rogue
Some clarifications because I feel like people might wonder why I picked one. Blood elves I specifically put with warlock because of the age old verdant spheres glyph and because in my mind, blood elves make the most sense as casters. Warlocks pretty much fit their social dynamic well I think.
Evoker and Dracthyr
Damn I’m so good at this 
Warrior = Orc
Paladin = Human
Hunter = BE or DS Troll only with axes instead of bow
Rogue = Orc maybe because of Garona
Priest = Human or Draenei
Shaman = Orc
Mage = Human
Warlock = Orc
Monk = Pandaren
Druid = NE
Demon Hunter = NE
Death Knight = Human
Evoker = Dracthyr
Engineering = Gnome
Blacksmithing = Goblin (?)
Alchemy = Forsaken
Leatherworking =
Herbalism = Tauren
Jewelcrafting =
Inscription =
Tailoring =
Enchanting =
Mining = Goblin or Dwarf
Skinning =
Warrior: Orc
Rogue: Undead
Mage: Blood Elf
Warlock: Orc or Undead
Shaman: Orc or Troll
Hunter: Dwarf or Blood Elf
Paladin: Draenei
Priest: Human
Demon Hunter: Night Elf
Druid: Night Elf
Death Knight: Human or Orc.
Monk: Pandaren
Evoker: Dead.
Warrior panda
Warlock panda
Demon hunter panda
Evoker panda
Druid panda
Monk panda
Hunter panda
Priest panda
Paladin panda
Rogue panda
Shaman panda
Death knight panda
Mage panda
Warrior - orc
Shaman - orc
Hunter - orc
Rogue - orc
Warlock - orc
Death Knight - orc
Mage - orc
Priest - orc
Druid - orc
Paladin - orc
Monk - orc
Demon hunter - orc
Evoked - orc
Goblins and Gnomes are distinctly lacking a class that fits them.
Both Races are built entirely around Technology, but we don’t have a Technology class.
So Goblin / Gnome. = Tinker.
I say this, but I have more Goblins than any other race, lol.
Peon - Orc
If it weren’t for the sheer Phoenix iconography Fire mages have I’d agree.
I agree but Goblins fit all three Rogue aspects well. We see them as rogues for Shadow and Assassin and they’re prominently shown as Pirates. Especially with a few engineering abilities thrown in. Gnomes/Goblins work well with the Hunter but Survival just doesn’t link up well.
I think people are forgetting one spec for Rogue is pirate like. I would have put Human/Kul Tiran up there too but just felt Humans really are represented well in all classes or at least certain specs.
The Burning Blade “Fireseers” fit the Phoenix, though I’d see it more shaman-esque.
If I had to come up with solutions for Classes that Fit Goblin and Gnome that aren’t Tinker… 
So Goblin Rogue makes sense, assuming you’re a sneak-thief or an enforcer for a Cartel. Subtley Rogue in particular works really good because they use Shadow, and Goblins are some of the most prevalent Twilights Hammer cultist / Old god Enthusiasts. (It was the Goblins who stamped the Elemental Plates into Deathwing’s body after all)
On this same vein, Goblin Shadow Priest would also work.
Gnome I really Struggle with. Most significant gnome characters are inventors, Tinkers, or Geniuses of their own design. The Significant Outliers like Millhouse Manastorm or Wilfred Fizzlebang are really just meme characters and don’t represent gnomish culture as a whole.
So While you could make a Gnome Mage or a Gnome Warlock and pantomime that meme, You can’t really make a Gnome of a class that represents Gnomes, in the same way you can make an Orc Shaman or a Night elf Druid. You follow? Gnomes are Great Tinkers and Inventors.