Car wash is the only legal way to use soap, for better or worse.
And doing what with it, exactly?
The trapped oil gets disposed of separately, the treated oil-free water goes to the sewer.
AFAIK the regulations were put into place to prevent untreated water from going into storm drains and into the water system, so presumably the cities are fine if you pour it down into the bathtub drain and let it go to the waste water treatment plants.
But car washes are interesting systems and one of the things that makes them interesting is that the water they use is (often) recycled. The same water gets reclaimed and re-used to wash cars over and over again.
I didn't look very deep at all, but one car wash chain suggests[1] a reclamation rate of 70%. Not perfect, but way better than 0.
Given that by hand you use much more hot water than the dishwasher would, you're using more electricity (or gas). Since a little pump doesn't use much at all, it's mainly just the drying heat that matters. Though nobody is forced to use the drying cycle - if one enjoys drying dishes by hand, they can still use the dishwasher to do the washing.
Is it safe? Not for a restaurant where the temps need to get to parasite killing temps to pass muster.. but if I had to guess, the odds of getting killed by a bad-tempered cow outweigh it by 10x.
eg. Our household conserves water (we track every litre we bring up to the tanks and filter) and we don't have a dishwasher - hot water is heated by the sun and run-off goes to the grey water tank for the garden.
You save water and electricity.