The Dog Island!

Azure Striker Gunvolt 3
Cave Story
Kingdom Hearts (goofy is a dog right)

Shadow Dancer: The Secret of Shinobi, the protagonist have a dog

Pom Gets Wifi (You play as a dog)
https://www.backloggd.com/games/pom-gets-wi-fi

Sonic Forces (You can make a dog character)
https://www.backloggd.com/games/sonic-forces

Megaman 3 (The first game were you can use rush to help you.
https://www.backloggd.com/games/mega-man-3/

Any Harvest Moon games and Stardew Valley (You have a dog pet)
https://www.backloggd.com/games/harvest-moon
https://www.backloggd.com/games/stardew-valley

Super Bomberman 5 (One of the main villain is a dog)
https://www.backloggd.com/games/super-bomberman-5/

Star Fox 2 (One of the main characters Fay is a female dog)
https://www.backloggd.com/games/star-fox-2/

Secret of Evermore (The protagonist have a dog which is his party member)
https://www.backloggd.com/games/secret-of-evermore

Gitaroo Man (The main character have a dog named Puma who gives him his guitar to fight)
https://www.backloggd.com/games/gitaroo-man/

The Sims 2 Pets (The game envolves about having pets, dogs included :D)
https://www.backloggd.com/games/the-sims-2-pets

Snoopy Concert (Its a click game for snes where you can play as snoopy and other characters)
https://www.backloggd.com/games/snoopy-concert/

Altered Beast (One of the main protagonist transformation is a wolf, wolfs are dogs right?)
https://www.backloggd.com/games/altered-beast

Earthworm Jim (There is a dog character named Pete and one of the stages envolves about protecting him)
https://www.backloggd.com/games/earthworm-jim

Persona 3 (One of the main party members is a dog named Koromaru)
https://www.backloggd.com/games/persona-3-fes

Added all of the suggestions!! :0

How on Earth did nobody think of Nintendogs before? XD

Also, if werewolves are valid (saw those Wolf Among Us and Altered Beast) you become one in the very first cutscene of Sonic Unleashed. =P

I don't think that technically counts as it's a werehog rather than a werewolf sorry