This is a little old but I have things to say about this since I used to write a little bit of fanfic.
Fanfic can teach you the writing style of manga, so in that you can learn how to write a manga-style romance, comedy, battle, etc. better if you ever want to make your own story, it can help you try to improve your writing if you're serious about the fanfic, and maybe some more gains but I only got about 2.
Writing a fanfic though really doesn't matter upon writing skill sometimes, just having read or watched the actual manga/anime would have given you a base to work on, how characters speak, act, the humor, the serious parts, the romance and everything so it would be easy in the aspect for sure.
@Maxite: I'm interested in becoming a proper writer even though I wrote a few fanfics, I know I have to improve my writing form and word choice, and though I did write fanfic for the love of the series, I still did it with joy since I enjoy creating stories anyways, it all depends on the individual, I happen to say this line a lot though.