Games like Project Songbird
Games like Project Songbird include Mouthwashing, SILENT HILL 2, Amnesia: A Machine For Pigs, SOMA, and SILENT HILL f. We matched them by the tags players give Project Songbird on Steam (Psychological Horror, Story Rich, Cinematic, Psychological, and FPS): closest matches first, and among equally close ones the better-known games.

Mouthwashing
Shared: Psychological Horror · Story Rich · Psychological

SILENT HILL 2
Shared: Psychological Horror · Story Rich · Psychological

Amnesia: A Machine For Pigs
Shared: Story Rich · Psychological Horror · Walking Simulator

SOMA
Shared: Psychological Horror · Story Rich · Atmospheric

SILENT HILL f
Shared: Psychological Horror · Story Rich · Horror

Layers of Fear (2016)
Shared: Psychological Horror · Story Rich · Walking Simulator

METRO 2033 Redux
Shared: Story Rich · FPS · Atmospheric

The Beginner's Guide
Shared: Story Rich · Walking Simulator · Atmospheric

MiSide
Shared: Psychological Horror · Story Rich · Horror

What Remains of Edith Finch
Shared: Story Rich · Walking Simulator · Atmospheric

Firewatch
Shared: Story Rich · Atmospheric · Walking Simulator

Still Wakes the Deep
Shared: Psychological Horror · Story Rich · Horror

KARMA: The Dark World
Shared: Psychological Horror · Walking Simulator · Story Rich

Metro Exodus
Shared: Story Rich · FPS · Atmospheric

METRO Last Light Redux
Shared: Story Rich · FPS · Atmospheric

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

Summer of '58
Shared: Psychological Horror · Cinematic · Atmospheric

Blair Witch
Shared: Psychological Horror · Horror · Story Rich

Resident Evil Village
Shared: Story Rich · Horror · First-Person

Outlast
Shared: Psychological Horror · Horror · Atmospheric

Fears to Fathom - Ironbark Lookout
Shared: Psychological Horror · Atmospheric · Horror

Half-Life: Alyx
Shared: Story Rich · FPS · Horror

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

Fran Bow
Shared: Psychological Horror · Story Rich · Horror