Games like Stay: Your Forever Pet!
Games like Stay: Your Forever Pet! include Disney Dreamlight Valley, Stardew Valley, Slime Rancher, Potion Craft: Alchemist Simulator, and Wobbledogs. We matched them by the tags players give Stay: Your Forever Pet! on Steam (Dogs, Exploration, Relaxing, Singleplayer, and Simulation): closest matches first, and among equally close ones the better-known games.

Disney Dreamlight Valley
Shared: Relaxing · Exploration · Life Sim

Stardew Valley
Shared: Relaxing · Life Sim · Simulation

Slime Rancher
Shared: Exploration · Singleplayer · Cute

Potion Craft: Alchemist Simulator
Shared: Relaxing · Singleplayer · Simulation

Wobbledogs
Shared: Dogs · Wholesome · Simulation

The Sims™ 4
Shared: Life Sim · Simulation · Singleplayer

Paralives
Shared: Life Sim · Simulation · Singleplayer

Crime Scene Cleaner
Shared: Simulation · Relaxing · Singleplayer

Dressmaker
Shared: Relaxing · Simulation · Singleplayer

Palia
Shared: Exploration · Life Sim · Relaxing

Slime Rancher 2
Shared: Relaxing · Singleplayer · Cute

A Short Hike
Shared: Exploration · Relaxing · Singleplayer

Unpacking
Shared: Relaxing · Singleplayer · Simulation

Hello Kitty Island Adventure
Shared: Exploration · Simulation · Life Sim

DAVE THE DIVER
Shared: Relaxing · Exploration · Singleplayer

My Time at Portia
Shared: Life Sim · Simulation · Singleplayer

Restore Your Island
Shared: Exploration · Relaxing · Simulation

Sticky Business
Shared: Relaxing · Simulation · Wholesome

Teddy's Haven
Shared: Relaxing · Simulation · Exploration

LOST EMBER
Shared: Exploration · Singleplayer · Relaxing

Coral Island
Shared: Relaxing · Life Sim · Simulation

Winter Burrow
Shared: Exploration · Crafting · Relaxing

Animal Shelter
Shared: Dogs · Life Sim · Singleplayer

Summer in Mara
Shared: Relaxing · Exploration · Life Sim