Naruto
Naruto is one of the clearest recs because both are about a loud underdog training in a class/team system while carrying a huge dream. The exams, mentors, rivalries, and gradual mastery of power make the structure feel very familiar.
Black Clover
Black Clover shares the underdog protagonist, magic/power society, rival dynamic, squad structure, and constant push toward becoming the top figure in the world. It is more fantasy than superhero, but the emotional formula lines up closely with My Hero Academia.
One Punch Man
Both explore superhero society, public rankings, villains, and what heroism means, but from different angles. One Punch Man is satire while My Hero Academia is earnest, so it works well for fans who want the same hero-world setup with a more comedic edge.
Tiger & Bunny
Tiger & Bunny is a strong fit for the professional hero side: branded heroes, public approval, rankings, and the tension between saving people and performing for society. It is older and more adult-cast focused, but the hero-industry worldbuilding overlaps a lot.
Assassination Classroom
The link is the classroom ensemble: students with very different abilities grow under a strange but caring teacher. Assassination Classroom has less superhero action, but the mentor-student bond and class development hit a similar emotional note.