Earlier quoted context omitted.
The bike is in fact the more appropriate transport for moving a 200lb person than a 5000lb vehicle. Literally 95% of the energy being consumed is just to move the damned vehicle around, not to do anything productive with it.
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.
I shower once a day regardless. I combine my bike commute on the way in with train or bus depending on the route I feel like. No sweat that way. The ride back I will do on a bike and take a shower after. Gets the cardio requirement done at the same time as the commute so it's a two birds one stone thing.