The original idea behind a "sonic shower" is that it minimizes water use, not that it uses no water at all.
As even this idea isn't explicit, the basic reason for the "sonic" part may be that it intensifies the cleaning effect or makes the experience more pleasurable, without affecting water consumption at all.
After all, it should be fairly trivial to set water consumption of a conventional shower to zero, by perfectly recycling a volume of water, say, ten liters of it. The life support system of a starship must be capable of far greater feats of recycling than that, at high reliability that would make any attempts at minimizing water consumption through more vulnerable technologies such as "sonics" look pretty ridiculous in comparison.
The first mentioning of the phrase that I can recall is TMP, when the probe-Ilia first shows up. We hear the computer voice say "Intruder location: A sonic shower".
Ah, good spotting - that one isn't in the transcripts. (Odd wording, though... Why the indefinite article?)
There's no real limit for when sonic showers might have been introduced. Perhaps they were commonplace in the 1950s already, in the Trek universe? But they play no role in dialogue until TNG "The Game", and after that they go unmentioned until DS9/VOY.