Kanto
Agatha: Gengar
Blaine: Arcanine (RBYFRLG) Rapidash (GSCHGSS)
Green/Blue: the starter (RBYFRLG) none (GSCSMUSM) Pidgeot (HGSS)
Brock: Onix
Bruno: Machamp
Erika: Vileplume and Victreebel (RBYFRLG) Victreebel and Bellossom (GSC) Bellossom (HGSS) Vileplume (LGPE)
Giovanni: Rhydon (RBLGPE) Nidoking and Rhydon (Y) Rhyhorn (FRLG) Nidoqueen (HGSS) Mewtwo (USM)
Janine: Venomoth
Karen: Houndoom
Kiyo: Hitmonlee and Hitmonchan
Koga: Weezing (RBFRLG) Venomoth (YLGPE) Crobat (GSCHGSS)
Lance: Dragonite
Lorelei: Lapras
Misty: Starmie
Red: whatever you want (RBYFRLGLGPE) Pokemon #025 (GSCHGSS)
Sabrina: Alakazam
Lt. Surge: Raichu (RBYFRLG) Electabuzz (GSCHGSS)
Johto
Archer: Houndoom
Ariana: none (GSCHGSS) Murkrow (FRLG)
Bugsy: Scyther
Chuck: Poliwrath
Clair: Kingdra
Eusine: Electrode
Falkner: Pidgeotto
Jasmine: Steelix
Joey: the top percentage Rattata
Li: Hoothoot
Morty: Gengar
Petrel: Weezing
Proton: Golbat (GSC) Weezing (HGSS)
Pryce: Piloswine
Silver: starter
Whitney: Miltank
Will: Xatu
Hoenn
Aarune: Flygon
Anabel: Snorlax
Archie: Sharpedo (SEAS) Kyogre (USM)
Brawly: Makuhita
Brendan/May: starter
Drake: Salamence
Flannery: Torkoal
Glacia: Walrein
Greta: Umbreon
Juan: Kingdra
Lucy: Seviper
Maxie: Camerupt (REOR) Groudon (USM)
Matt: none (RSE) Sharpedo (ORAS)
Norman: Slaking
Phoebe: Dusclops (RSE) Dusknoir (ORAS)
Shelley: none (RSE) Sharpedo (ORAS)
Sidney: Absol
Spenser: Slaking
Steven: Metagross
Tabitha: none (RS) Golbat (E) Camerupt (ORAS)
Courtney: none (RS) Camerupt (ORAS)
Tate and Liza: Solrock and Lunatone
Tucker: Swampert
Wallace: Milotic
Wally: Gardevoir (RSE) Gallade (ORAS)
Wattson: Magneton (RSORAS) Manectric (E)
Zinnia: Salamence
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
source: a lot of Bulbapedia pages
