Games like Rescue the Enchanter
Games like Rescue the Enchanter include Storyteller, Big Walk, The Witness, There Is No Game: Wrong Dimension, and Is This Seat Taken?. We matched them by the tags players give Rescue the Enchanter on Steam (Exploration, Interactive Fiction, Puzzle, Point & Click, and 2D): closest matches first, and among equally close ones the better-known games.

Storyteller
Shared: Puzzle · Interactive Fiction · Point & Click

Big Walk
Shared: Exploration · Puzzle · Relaxing

The Witness
Shared: Exploration · Puzzle · First-Person

There Is No Game: Wrong Dimension
Shared: Puzzle · Point & Click · Casual

Is This Seat Taken?
Shared: Puzzle · Casual · Relaxing

Slime Rancher
Shared: Exploration · First-Person · Colorful

A Little to the Left
Shared: Puzzle · Relaxing · Point & Click

Samorost 1
Shared: Point & Click · Puzzle · Casual

Chants of Sennaar
Shared: Puzzle · Point & Click · Colorful

Morphopolis
Shared: Puzzle · Point & Click · Casual

Monument Valley
Shared: Puzzle · Exploration · Relaxing

Gorogoa
Shared: Puzzle · Point & Click · Casual

The Room 4: Old Sins
Shared: Puzzle · Point & Click · Exploration

The Room
Shared: Puzzle · Point & Click · First-Person

Blue Prince
Shared: Exploration · Puzzle · First-Person

Carto
Shared: Exploration · Puzzle · Relaxing

Strange Jigsaws
Shared: Puzzle · Casual · 2D

Zup! X
Shared: Puzzle · Casual · 2D

FEZ
Shared: Puzzle · Exploration · Casual

Unboxing the Cryptic Killer
Shared: Puzzle · Point & Click · Exploration

Zup! S
Shared: Puzzle · Casual · 2D

Lost in Play
Shared: Puzzle · Point & Click · 2D

Murders on the Yangtze River
Shared: Puzzle · Exploration · Adventure

Doors: Paradox
Shared: Puzzle · Point & Click · Relaxing