Your relative masses are obviously exaggerated.

Showering uses hot water and you also must change clothes, which also require cleaning. If you are doing this twice per day in addition to someone using some other means of transportation there is a non trivial energy cost involved.

If the car propulsion is non fossil fuel based then the car wins because you are using much less water.

Showering does use hot water, but, it's maybe 20-30 litres, and you're heating up by what 25-30 K ? That's just not very much energy, and since we want heat we can go via a heat pump to do less work, whereas that's not an option for the car.

I did some envelope guesses and I can't see how this can come out for the car.