The best family games like Ape Escape are Ape Escape 2, Pikmin and Psychonauts. Below are 5, each with the reason it is here — and, at the end, what filtering to family costs you.
Filtering to family means the game has to work for a child and an adult at the same table without either of them being bored. That is a much harder design problem than a difficulty slider.
What you get: Something you can put in front of anyone. The best family games make the skill gap not matter rather than pretending it is not there.
What made Ape Escape worth playing in the first place is covered on the full list. This page is the family part of that answer, and only the family part.
More of exactly the same, with the gadget set expanded.
The other Sony-era-adjacent game about catching and herding small creatures.
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.
Edge. Family games rarely go anywhere uncomfortable, and a few of them are too generous to hold an experienced player for long alone.
Specifically, these are strong answers for Ape Escape that are not family 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.
Age ratings cover content but not difficulty or frustration, which is what actually ruins a family session. Five minutes of unedited play footage tells you more than any rating.
With mixed ages, co-operative usually wins. The competitive ones that work are the ones where being in last place is still entertaining.
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 family — 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 family guide for where to start in the genre generally.