Manga in which the protagonist can enter stories
It could be KINGDOM HEARTS, and is actually based on a PlayStation 2 game that came out first.

In the main story, Sora (the protagonist) travels among other worlds. And in each world, he transforms and blends in to resemble the world's characters.
Also in every world, there is a problem that only he could help to fix.

In the first season of this series, there are 43 chapters