Fan fiction is loosely defined as written works that feature characters and elements from a TV series, a film, or a book. Without fan fiction we wouldn’t have works like “Pride and Prejudice and Zombies.”
Should fan fiction be a legitimate genre for your English classes?
In fact, there already are published works of literature that technically fit the category of fan fiction.
“Grendel” by John Gardner is an interpretation of “Beowülf” from the perspective of the monster Grendel.

Dante’s “Divine Comedy” and “Paradise Lost” by John Milton can be interpreted as fan fiction of the Bible.

Neil Gaiman wrote “Snow, Glass, and Apples,” which explores the tale of Snow White from the queen’s eyes.

“Wide Sargasso Sea” by Jean Rhys is a 1966 prequel to Charlotte Brontë’s “Jane Eyre.”

Literature (on Merriam-Webster) is defined as “written works (such as poems, plays, and novels) that are considered to be very good and to have lasting importance: books, articles, etc., about a particular subject; printed materials (such as booklets, leaflets, and brochures) that provide information about something.”
The first and second definitions apply to fan fiction. Some argue against it because this type of writing doesn’t fit their definitions of art, or don’t have “lasting merit.” That’s really a matter of opinion.