Does radiative recombination occur in regular diodes as well?
Radiative recombination is generally not significant in silicon diodes because silicon is an indirect bandgap semiconductor. This means that the highest energy states in the valence band and the lowest energy states in the conduction band have different momentum, so there's no way for an electron to transfer directly from one to the other while also conserving momentum. 

Recombination in silicon therefore generally involves intermediate states associated with impurities (this is called Shockley-Read-Hall recombination after three guys who first described it in the literature). Since the impurity sites are localized in space, they are spread out in momentum (due to Heisenberg's principle), and they enable the captured electron to transfer momentum to or from the crystal lattice.