This discussion will be expanded and made quantitative. But the essence of it is the following:
Sound waves are tied to the medium (usually air) and we measure both the velocity of the observer and the velocity of the source relative to that medium. Motion of either the source or the observer relative to the air will give a Doppler Shift and the formula contains both velocities.
Light waves and any EM waves for that matter have no medium. We cannot speak of a velocity for the source and a velocity for the observer separately. There is only the relative velocity of the two. They are approaching or separating at speed v.
And with that we cannot forget what is possibly the corniest joke in physics. You see an optical Doppler Shift (a red shift) every night when you drive your car, right? The light from the cars coming at you looks white, but the light from the ones going away looks red.
Ouch. Just Ouch.