Games like STAY
Games like STAY include OneShot, The Test, No, I'm not a Human, Little Nightmares, and Little Nightmares II. We matched them by the tags players give STAY on Steam (Indie, Adventure, Choices Matter, Story Rich, and Puzzle): closest matches first, and among equally close ones the better-known games.

OneShot
Shared: Indie · Story Rich · Pixel Graphics

The Test
Shared: Story Rich · Pixel Graphics · Choose Your Own Adventure

No, I'm not a Human
Shared: Indie · Choices Matter · Horror

Little Nightmares
Shared: Adventure · Indie · Horror

Little Nightmares II
Shared: Adventure · Puzzle · Horror

Sally Face - Episode One
Shared: Indie · Story Rich · Adventure

DISTRAINT: Deluxe Edition
Shared: Indie · Adventure · Pixel Graphics

INSIDE
Shared: Indie · Story Rich · Adventure

INMOST
Shared: Indie · Adventure · Pixel Graphics

Undertale
Shared: Choices Matter · Story Rich · Indie

Detention
Shared: Indie · Adventure · Horror

Tales of the Black Forest
Shared: Indie · Adventure · Pixel Graphics

Fran Bow
Shared: Story Rich · Adventure · Indie

Oxenfree
Shared: Choices Matter · Story Rich · Indie

Late Shift
Shared: Choices Matter · Adventure · Indie

Tiny Bunny
Shared: Story Rich · Choices Matter · Horror

OMORI
Shared: Indie · Story Rich · Pixel Graphics

Roadwarden
Shared: Adventure · Indie · Pixel Graphics

To the Moon
Shared: Indie · Story Rich · Adventure

Little Misfortune
Shared: Indie · Story Rich · Choices Matter

Replica
Shared: Indie · Adventure · Puzzle

The Wolf Among Us
Shared: Adventure · Story Rich · Choices Matter

The Red Strings Club
Shared: Choices Matter · Story Rich · Pixel Graphics

Papers, Please
Shared: Indie · Pixel Graphics · Puzzle