The best platformer games like Ape Escape are Ape Escape 2, Jak and Daxter and Psychonauts. Below are 5, 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 Ape Escape 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.
More of exactly the same, with the gadget set expanded.
The PS2 platformer that inherited Ape Escape's collect-everything structure.
A gadget-driven 3D platformer with far stranger levels.
Capture mechanics turn every enemy into a new gadget.
Ability-swapping platforming that keeps inventing new verbs.
Weight. Platformers rarely carry a story you will think about afterwards, and the collectible-led ones can become checklists if you let them.
Specifically, these are strong answers for Ape Escape that are not platformer games, so they are not on this page:
The full list is on games like Ape Escape — 9 picks, no genre filter.
Ape Escape 2. More of exactly the same, with the gadget set expanded.
No. Ape Escape released on the PlayStation, 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 Ape Escape 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.