Adventures of Mana is the second remake of a GameBoy title, Seiken Densetsu: Final Fantasy Gaiden (known as Final Fantasy Adventure in North America and Mystic Quest in Europe), a top-down action-adventure game with a few rpg elements, reminiscent of the first The Legend of Zelda. Despite being 3D modeled, the world is structured the same way of the original, being made up of multiple “zones” like Hyrule in the first adventure of Link, and the progression is likewise linear and dipends on puzzles, key items and boss battles. The game is not the greatest ever sadly, has many gameplay issues, weird glitches and the visuals are quite cheap but not considering mobile, it's a PS Vita exclusive, and the start of the 3D remaking of the Mana series - followed by Secret of Mana and Trials of Mana (Seiken Densetsu 3).

Oceanhorn: Monster of Uncharted Seas shares many similarities with The Legend of Zelda (Wind Waker) as well but is on Switch too, and personally I'm not a fan.

There are many other action/adventure games on PS Vita, but nothing else quite close to the Nintendo series, most are also rpgs like the Ys series, that is known for its fast combat. The structure is very similar to The Legend of Zelda though, again with puzzles, key items (with some backtracking too) and boss battles, if you wanted to give it a try I'd recommend Memories of Celceta (not on Switch).

Many have already suggested a lot of great rhythm games, so to those I'll add Sound Shapes, a in-house Sony production that's more of a 2D platformer, where you can play or compose tracks interacting with or adding tunes whose pitch and instrument depend on the disposition in the actual level. Another alternative rhythm-based game is Bit.Trip Presents...Runner2: Future Legend of Rhythm Alien, an auto-running platformer inspired by the Bit.Trip series. Finally there's also the classic Patapon PSP games.

Oh and on Switch there's Mad Rat Dead, yet another rhythm-based 2D platformer with an amazing sountrack, check it out!