Some cities have combined or separated sewer systems. Even if combined, it may be designed to overflow during heavy rain, so it's not a guarantee that car wash water with dirt, soap and oil will not go into into a stream somewhere although in that case you're also sending literal shit there. Also when combined, there may still be old infrastructure that drains to a stream or river so a blanket ban is a good idea.
Typically a car wash would be required to have an oil-water separator (with maintenance records and occasional checks) and discharge effluent to the sanitary sewer. Not sure about everywhere but in Vancouver (I have experience working in water treatment there) you also need to have the car wash covered and send collected rainwater to the storm sewer.
Perhaps there could be a middle ground where you're allowed to wash in your driveway but only with a specific soap, and not allowed to degrease your engine bay. There's basically no way to enforce that though,.
Also might as well note here that in Vancouver storm drains that connect to the storm sewer have little fish stenciled by them.