Games like Cat
Games like Cat include Banana, Bongo Cat, Nodebuster, Shelldiver, and That's not my Neighbor. We matched them by the tags players give Cat on Steam (Casual, Simulation, Incremental, Point & Click, and Arcade): closest matches first, and among equally close ones the better-known games.

Banana
Shared: Incremental · Casual · 2D

Bongo Cat
Shared: Cats · Casual · Free to Play

Nodebuster
Shared: Casual · Incremental · Simulation

Shelldiver
Shared: Casual · Incremental · Arcade

That's not my Neighbor
Shared: Casual · Point & Click · Simulation

Digseum
Shared: Casual · Incremental · Simulation

Click and Conquer
Shared: Casual · Incremental · Simulation

Cookie Clicker
Shared: Incremental · Casual · Horror

A Game About Feeding A Black Hole
Shared: Casual · Incremental · Simulation

VHS
Shared: Casual · Simulation · Incremental

Fish
Shared: Casual · Incremental · Simulation

Pinball FX Classic
Shared: Simulation · Casual · Arcade

Poop Clicker
Shared: Casual · Simulation · Point & Click

Click the Button
Shared: Casual · Incremental · Simulation

Insaniquarium Deluxe
Shared: Casual · Incremental · Simulation

Russian Life Simulator
Shared: Simulation · Incremental · 2D

The Sims™ 4
Shared: Simulation · Free to Play · Casual

100 Korea Cats
Shared: Casual · Cats · Point & Click

To The Core
Shared: Casual · Incremental · Arcade

Travellin Cats in Paris
Shared: Cats · Casual · Free to Play

Game Dev Tycoon
Shared: Simulation · Casual · 2D

100 Christmas Cats
Shared: Cats · Casual · Free to Play

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

Shower With Your Dad Simulator 2015: Do You Still Shower With Your Dad
Shared: Casual · Arcade · 2D