The best indie games like Ico are Brothers: A Tale of Two Sons, Journey and Rime. 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 Ico 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.
Two characters on one controller — the companion is a control scheme, not an escort.
Wordless companionship with a stranger, and the same restraint.
Openly Ico-inspired: an island, a boy, environmental puzzles, no dialogue.
Silent, bleak, and every puzzle explains the world instead of a tutorial doing it.
A companion who is also the obstacle, in a game about something harder than it looks.
Atmosphere and scale doing the work that dialogue usually does.
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 Ico that are not indie games, so they are not on this page:
The full list is on games like Ico — 9 picks, no genre filter.
Brothers: A Tale of Two Sons. Two characters on one controller — the companion is a control scheme, not an escort.
No. Ico released on the PlayStation 2, 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 Ico 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.