Games like The Puppy
Games like The Puppy include Five Nights at Freddy's, Little Misfortune, That's not my Neighbor, Five Nights at Freddy's: Into the Pit, and Fran Bow. We matched them by the tags players give The Puppy on Steam (Adventure, Point & Click, Time Management, 2D, and Horror): closest matches first, and among equally close ones the better-known games.

Five Nights at Freddy's
Shared: Horror · Survival Horror · Point & Click

Little Misfortune
Shared: Point & Click · Adventure · Indie

That's not my Neighbor
Shared: Adventure · Point & Click · Horror

Five Nights at Freddy's: Into the Pit
Shared: Adventure · Horror · 2D

Fran Bow
Shared: Point & Click · Adventure · Horror

Lethal Company
Shared: Horror · Survival Horror · PvE

The Coma 2: Vicious Sisters
Shared: Adventure · Horror · Survival Horror

Rusty Lake Paradise
Shared: Point & Click · Adventure · Horror

The Forest
Shared: Adventure · Horror · Survival Horror

Papers, Please
Shared: Point & Click · Indie · Simulation

There Is No Game: Wrong Dimension
Shared: Point & Click · Adventure · Indie

American Arcadia
Shared: Point & Click · 2D · Horror

Sally Face - Episode One
Shared: Adventure · Horror · Indie

The Cat Lady
Shared: Adventure · Horror · Point & Click

Samsara Room
Shared: Adventure · Point & Click · Indie

Poppy Playtime
Shared: Horror · Survival Horror · Adventure

Little Nightmares
Shared: Adventure · Horror · Indie

DreadOut
Shared: Adventure · Horror · Indie

Five Nights at Freddy's: Security Breach
Shared: Horror · Survival Horror · Adventure

Deponia
Shared: Adventure · Point & Click · 2D

Servant of the Lake
Shared: Adventure · Point & Click · Indie

The Witch's House MV
Shared: Horror · 2D · Adventure

Look Outside
Shared: Adventure · Horror · Survival Horror

Hollow Knight: Silksong
Shared: 2D · Indie · Adventure