Quoted By:
Sinnoh
Aaron: Drapion
Barry: starter
Bertha: Hippowdon (DP) Rhyperior (Pt)
Buck: Claydol
Byron: Bastiodon
Candice: Abomasnow (DP) Froslass (Pt)
Cheryl: Chansey
Cynthia: Garchomp
Cyrus: Weavile (DPPt) Dialga (US) Palkia (UM)
Darach: Empoleon
Fantina: Mismagius
Flint: Infernape (DP) Magmortar (Pt)
Gardeina: Roserade
Jupiter: Skuntank
Lucas/Dawn: starter
Lucian: Bronzong (DP) Gallade (Pt)
Marley: Arcanine
Mars: Purugly
Mira: Kadabra
Maylene: Lucario
Riley: Lucario
Roark: Cranidos
Saturn: Toxicroak
Volkner: Luxray (DP) Electivire (Pt)
Unova
Alder: Volcarona
Benga: Volcarona
Bianca: starter
Brycen: Beartic
Burgh: Leavanny
Caitlin: Gothitelle
Cheren: starter
Chili: Pansear
Cilan: Pansage
Clay: Excadrill
Colress: Klinklang
Cress: Panpour
Drayden/Iris: Haxorus (Iris never used Axew in the core series games)
Elesa: Zebstrika
Ghetsis: Hydreigon (BWBW2) Zekrom (US) Reshiram (UM)
Grimsley: Bisharp
Hugh: starter
Lenora: Watchog
Marshal: Mienshao (BW) Conkeldurr (BW2)
Marlon: Jellicent
Rosa/Nate: starter
Roxie: Whirlipede
Shadow Triad: Absol, Banette, Accelgor
Shauntal: Chandelure
Skyla: Swanna
Zinzolin: Weavile
Kalos
Aliana: Druddigon
Bryony: Bisharp
Calem/Serena: starter
Celosia: Drapion
Clemont: Heliolisk
Dexio: none (XY) Alakazam (SMUSM)
Diantha: Gardevoir
Drasna: Noivern
Emma: Malamar
Grant: Amaura and Tyrunt
Lysandre: Gyarados (XY) Xerneas (US) Yveltal (UM)
Malva: Talonflame
Olympia: Meowstic
Ramos: Gogoat
Shauna: starter
Siebold: Barbarnacle
Sina: none (XY) Glaceon (MUM)
Tierno: Crawdaunt
Trevor: Florges
Valerie: Sylveon
Viola: Vivillon
Wikstrom: Aegislash
Wulfric: Avalugg
Xerosic: Malamar
I'm not going to do Alola because this question was asked in 2013 and I'm lazy.
Real life
Morimoto: Zebstrika (BWBW2) none (SMUSM)
Nishino: Snorlax