Games like House of Rules
Games like House of Rules include Little Nightmares, Sally Face - Episode One, Little Nightmares II, No, I'm not a Human, and OneShot. We matched them by the tags players give House of Rules on Steam (Puzzle, Multiple Endings, Female Protagonist, Story Rich, and Mystery): closest matches first, and among equally close ones the better-known games.

Little Nightmares
Shared: Puzzle · Adventure · Horror

Sally Face - Episode One
Shared: Story Rich · Puzzle · Psychological Horror

Little Nightmares II
Shared: Puzzle · Adventure · Horror

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

OneShot
Shared: Story Rich · Puzzle · Multiple Endings

Oxenfree
Shared: Story Rich · Female Protagonist · Choices Matter

INSIDE
Shared: Story Rich · Puzzle Platformer · Horror

What Remains of Edith Finch
Shared: Story Rich · Mystery · Female Protagonist

Fran Bow
Shared: Puzzle · Story Rich · Psychological Horror

The Room Three
Shared: Puzzle · Mystery · Adventure

Poppy Playtime
Shared: Puzzle · Horror · Mystery

Zero Escape: The Nonary Games
Shared: Puzzle · Story Rich · Multiple Endings

Outer Wilds
Shared: Story Rich · Mystery · Puzzle

Little Misfortune
Shared: Story Rich · Psychological Horror · Female Protagonist

Return of the Obra Dinn
Shared: Puzzle · Mystery · Story Rich

Cube Escape Collection
Shared: Puzzle · Mystery · Story Rich

Firework
Shared: Puzzle · Mystery · Horror

Detention
Shared: Puzzle · Story Rich · Psychological Horror

Life is Strange: Before the Storm
Shared: Story Rich · Female Protagonist · Choices Matter

The Test
Shared: Multiple Endings · Story Rich · Psychological Horror

Undertale
Shared: Multiple Endings · Story Rich · Choices Matter

Zero Escape: Zero Time Dilemma
Shared: Puzzle · Story Rich · Mystery

OMORI
Shared: Story Rich · Psychological Horror · Multiple Endings

DISTRAINT: Deluxe Edition
Shared: Psychological Horror · Story Rich · Adventure