No, you play a human in Fae Farm, who gets stranded in fairy land. You can get your character wings and kinda do a gliding jump that looks fairy like, I guess. But that's it.
The only games I know where you can explicitly play as a fairy are The Sims 3 with Supernatural DLC and "Faery: Legends of Avalon" (not cozy).
Personally, which game gave me the feeling of being a fairy the most is Smalland!
You play as an individual of a tiny folk who calls themselves "Smallfolk". And humans are called Giants. You can ride lady bugs and insects in general are huge creatures. You build your house out of leafs and stuff. And you can also get wings and glide.
It's a survival builder, though. In Early Access on Steam.
Just as a hint if you happen to be a fairy fan like me, who is desperate at the lack of games which allow you to be a fairy lol