The best indie games like Concrete Genie are Chicory: A Colorful Tale, Gris and A Short Hike. Below are 6, each with the reason it is here — and, at the end, what filtering to indie costs you.
Filtering to indie means you want the answer a small team would give. It is usually the most interesting filter on this site, because small studios take the risks large ones cannot.
What you get: One strong idea, explored completely, at a price that makes trying it easy. Indies have no production to hide behind, so the design is exposed.
What made Concrete Genie worth playing in the first place is covered on the full list. This page is the indie part of that answer, and only the indie part.
Painting the world as the whole mechanic, and it handles its themes beautifully.
Colour returning to a world as the story, wordlessly.
Small, warm and finishable in one sitting.
Short, wordless, and built on feeling rather than challenge.
Gentle, and unafraid of sad subject matter.
No combat, a striking art style, and a world to wander.
Polish and scale, sometimes. A few of these are rough at the edges, and a big-budget pick will always look better doing the same thing.
Specifically, these are strong answers for Concrete Genie that are not indie games, so they are not on this page:
The full list is on games like Concrete Genie — 9 picks, no genre filter.
Chicory: A Colorful Tale. Painting the world as the whole mechanic, and it handles its themes beautifully.
No. Concrete Genie released on the PlayStation 4, but a recommendation is only useful if you can play it, so the picks above span platforms and eras. What they share is the genre and the reason.
Self-published, or published without a large publisher's money. It says nothing about the size of the game, which is why a forty-hour metroidvania and a two-hour art piece sit under the same label.
Constraint. Without a budget to hide behind, one idea has to carry the whole thing — and a small team can commit to something risky that a large one would sand down.
By hand. Each pick was written for the full Concrete Genie list first, on what the game actually does, and this page is the part of that list that is indie — never a genre tag lookup, which is how "best racing games like Super Mario 64" used to be a page on this site.
Or read the indie guide for where to start in the genre generally.