Hey! I'm so glad you're enjoying the game :)
I don't usually like posting spoilers in the itch comments, but I'm sure if somebody doesn't want to be spoiled, they just won't keep reading this comment :p
The Beastial Steed will find you once you have met enough pets (: They don't need to be unique, you could just go somewhere that you can complete easily, and meet the same pet multiple times.