The first game I ever played of the genre was probably Wandering Willows. It’s quite old, release was in 2009. It was very different from all the other games I played up until then (which was mostly Final Fantasy, Zelda, Mortal Combat and the like xD).
You are stranded on an island and have to fulfill quests and help the villagers. As a companion you can catch various pets and level them. There is also farming and character customization.
My first cozy games were Animal Crossing on the GameCube and Harvest Moon it’s a wonderful life on GameCube. I fell in love with both games and played them endlessly.