Games like Copycat
Games like Copycat include Stray, Brothers - A Tale of Two Sons, Firewatch, What Remains of Edith Finch, and The Beginner's Guide. We matched them by the tags players give Copycat on Steam (Cats, Emotional, Indie, Adventure, and Story Rich): closest matches first, and among equally close ones the better-known games.

Stray
Shared: Cats · Adventure · Singleplayer

Brothers - A Tale of Two Sons
Shared: Adventure · Indie · Story Rich

Firewatch
Shared: Adventure · Story Rich · Atmospheric

What Remains of Edith Finch
Shared: Story Rich · Atmospheric · Adventure

The Beginner's Guide
Shared: Indie · Story Rich · Atmospheric

My Little Puppy
Shared: Emotional · Adventure · Story Rich

The Last Campfire
Shared: Adventure · Indie · Singleplayer

To the Moon
Shared: Indie · Emotional · Story Rich

Before Your Eyes
Shared: Emotional · Indie · Story Rich

Tomb Raider Game of the Year
Shared: Adventure · Story Rich · Singleplayer

Outer Wilds
Shared: Adventure · Story Rich · Exploration

A Plague Tale: Innocence
Shared: Adventure · Story Rich · Singleplayer

Hellblade: Senua's Sacrifice
Shared: Atmospheric · Story Rich · Singleplayer

Little Nightmares
Shared: Adventure · Atmospheric · Singleplayer

OneShot
Shared: Indie · Story Rich · Atmospheric

DEATH STRANDING DIRECTOR'S CUT
Shared: Story Rich · Atmospheric · Adventure

The Cat Lady
Shared: Adventure · Story Rich · Indie

Rise of the Tomb Raider™
Shared: Adventure · Singleplayer · Story Rich

Gone Home
Shared: Indie · Walking Simulator · Story Rich

The Witcher 3: Wild Hunt — Remastered
Shared: Story Rich · Adventure · Atmospheric

Hogwarts Legacy
Shared: Adventure · Singleplayer · Story Rich

Layers of Fear (2016)
Shared: Atmospheric · Indie · Singleplayer

GRIS
Shared: Indie · Atmospheric · Adventure

Finding Paradise
Shared: Emotional · Story Rich · Adventure