Games like The Insignia Project
Games like The Insignia Project include Lethal Company, METRO 2033 Redux, MiSide, Metro Exodus, and MOLE. We matched them by the tags players give The Insignia Project on Steam (Horror, Psychological Horror, Dystopian , Post-apocalyptic, and Alternate History): closest matches first, and among equally close ones the better-known games.

Lethal Company
Shared: Horror · Psychological Horror · First-Person

METRO 2033 Redux
Shared: Post-apocalyptic · Horror · FPS

MiSide
Shared: Psychological Horror · Horror · 3D

Metro Exodus
Shared: Post-apocalyptic · FPS · Horror

MOLE
Shared: Horror · Psychological Horror · Simulation

Inside the Backrooms
Shared: Horror · Psychological Horror · FPS

SOMA
Shared: Horror · Psychological Horror · Atmospheric

Escape the Backrooms
Shared: Horror · Psychological Horror · 3D

METRO Last Light Redux
Shared: Post-apocalyptic · Horror · FPS

Phasmophobia
Shared: Horror · Psychological Horror · First-Person

Outlast
Shared: Horror · Psychological Horror · Atmospheric

Iron Lung
Shared: Horror · Psychological Horror · Atmospheric

Backrooms: Escape Together
Shared: Horror · Psychological Horror · Atmospheric

BioShock™ Remastered
Shared: Horror · FPS · Dystopian

The Outlast Trials
Shared: Horror · Psychological Horror · First-Person

Mouthwashing
Shared: Horror · Psychological Horror · Atmospheric

Hamster Hunter
Shared: Psychological Horror · Horror · 3D

No, I'm not a Human
Shared: Horror · Psychological Horror · Atmospheric

Zort
Shared: Horror · Psychological Horror · Atmospheric

Species: Unknown
Shared: Horror · Atmospheric · First-Person

Outlast 2
Shared: Horror · Psychological Horror · Atmospheric

Five Nights at Freddy's
Shared: Horror · Psychological Horror · Indie

Buckshot Roulette
Shared: Horror · Psychological Horror · 3D

PILGRIM
Shared: Horror · 3D · Psychological Horror