@Mweeheehee
I think they can get around licensing by just using covers of the songs. Getting a cover license (in America at least) is really easy and the artist/estate cannot refuse according to this article: https://flypaper.soundfly.com/hustle/how-to-legally-cover-a-song/
No idea how cover rights work internationally (particularly Japan).
Honestly the only part where this would be a "problem" is in the montage where Shiori is listening to all that music to write her own song. MOST of the performances of famous music in the manga are not done by the original artists. Even when the spirit Jacks into the character, I'm pretty sure it's the character we hear singing.
However we still have the problem of writing believably "mind blowing music."
EDIT: Nevermind, I did more research, it's more complicated than that. Looks like it'll need more work even to "cover" the music in a anime.