For sound waves, the Doppler's formula for frequency shift differs slightly between the two situation :
(i) source at rest , observer moving (ii) source moving , observer at rest.
The exact Doppler formulae for the case of light waves in vacuum, are however, strictly identical for the two situations in case of light travelling in a medium ?

Explanation

No
Sound waves can propagate only through a medium. The two given situations are not scientifically identical because the motion of an observer relative to a medium is different in the two situations. Hence, the Dopper formulas for the two situations cannot be the same.
In case of light waves, sound can travel in a vacuum, the above two cases are identical because the speed of ligth is independent of the motion of the abserver and the motion of the source. When light travels , the above two cases are not identical because the speed of light depends on the wavelength of the medium.