As a carwash owner, I'm pretty versed on this law.
Technically, according to the Clean Water Act, nothing is allowed to enter the storm water management system except rainwater. This includes water run-off from washing your car. The effluent residue in the water from washing your car contains phosphates from the chemicals, and contaminates from the various parts of your car... rust, oil, gasoline, carbon, etc... The amount may be quite small from just your car, but imagine all the runoff from all the people who wash their cars in all of LA. This is part of the reason why santa monica bay is closed due to pollution after most large rainstorms.
Many cities and counties have incorporated the clean water act into their own rules, but most of them are not enforced. Ever hear of the car wash police? Me neither.
Mobile detailers are supposed to trap, suck up and then properly dispose of their wastewater. Most don't do this.
So, to answer your Q, it is not illegal to wash your car, but it is illegal to let the water run into the storm drain. Parking it on your lawn while you wash it is a solution. Or bring your car to a commercial car wash (like mine

) where the waste water is disposed of into the sewer system.
And the sanitary sewer system is a closed system. Those holes do not lead to the sewer.