We explain the existence of neutrino masses and their flavour structure, dark matter relic abundance and the observed 3.5 keV X-ray line within the framework of a gauged U(1) L-mu-L-tau extension of the "scotogenic" model. In the U(1) L-mu-L-tau symmetric limit, two of the RH neutrinos are degenerate in mass, while the third is heavier. The U(1) L-mu-L-tau symmetry is broken spontaneously. Firstly, this breaks the mu-tau symmetry in the light neutrino sector. Secondly, this results in mild splitting of the two degenerate RH neutrinos, with their mass difference given in terms of the U(1) L-mu-L-tau breaking parameter. Finally, we get a massive Z(mu tau) gauge boson. Due to the added Z(2) symmetry under which the RH neutrinos and the inert doublet are odd, the canonical Type-I seesaw is forbidden and the tiny neutrino masses are generated radiatively at one loop. The same Z(2) symmetry also ensures that the lightest RH neutrino is stable and the other two can only decay into the lightest one. This makes the two nearly-degenerate lighter neutrinos a two-component dark matter, which in our model are produced by the freeze-in mechanism via the decay of the Z(mu tau) gauge boson in the early universe. We show that the next-to-lightest RH neutrino has a very long lifetime and decays into the lightest one at the present epoch explaining the observed 3.5 keV line.