Does Yoshi have his own game

Yes! Yoshi debuted in a game called "Yoshi" on NES. The game is a cross between "Tetris" and "Bejeweled," so it's not really the kind of gameplay you're asking about.

After making his "official" debut in "Super Mario World," Yoshi went on to star in "Super Mario World 2: Yoshi's Island." In this game, you play as Yoshi (several different Yoshi's, actually), trying to rescue Baby Luigi from Kamek & Baby Bowser, all the while protecting Baby Mario from enemies. It's a very artistic game, and the levels are well thought out and challenging!

Yoshi's other playable appearances include:

Yoshi's Story (N64)
Yoshi's Universal Gravitation (GBA)
Yoshi Touch & Go (DS)
Yoshi's Island DS (DS)