In my view both are ephemeral as opposed to part of a continuing tradition.

Music Hall songs were composed or adapted folk/parlour songs for the pirpose of performance in Music Halls

Shanties were made as working songs for sailing vessels (though as I said mess hall songs could be seen as a continuation of forbitters). Fok singers may still be singing the songs (shanties), but if they are not sung any more as working songs, are thye really a continuation of this tradition?