The best platformer games like Yoshi's Island are Yoshi's Woolly World, Super Mario World and Kirby's Adventure. Below are 9, each with the reason it is here — and, at the end, what filtering to platformer costs you.
Filtering to platformer means you want the movement itself to be the pleasure. It is the right filter when the thing you liked was getting from one place to another.
What you get: A game you can judge in ten seconds. Platformers live or die on the jump, and a good one feels right before anything else happens.
What made Yoshi's Island worth playing in the first place is covered on the full list. This page is the platformer part of that answer, and only the platformer part.
The best of the modern entries, and the craft look is gorgeous.
The game it is a sequel to, and the best-designed 2D Mario.
The other Nintendo platformer built to be kind.
The other one that rewrites the genre's rules entirely.
The most polished 2D platformer since.
The SNES platformer with the best secret design.
The modern 2D Mario, and the closest in playfulness.
If the handmade art style was the reason.
3D platforming built to be generous and full of small inventions.
Weight. Platformers rarely carry a story you will think about afterwards, and the collectible-led ones can become checklists if you let them.
The full list is on games like Yoshi's Island — 9 picks, no genre filter.
Yoshi's Woolly World. The best of the modern entries, and the craft look is gorgeous.
No. Yoshi's Island released on the Super Nintendo, 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.
Whether the move set is complete early. The great ones give you everything in the first level and spend the rest asking better questions of it; the weak ones drip-feed abilities to disguise thin level design.
Different games. 2D platformers are about precision on a single screen; 3D ones are about reading space and camera. Each pick below says which it is.
By hand. Each pick was written for the full Yoshi's Island list first, on what the game actually does, and this page is the part of that list that is platformer — 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 platformer guide for where to start in the genre generally.