During my many years of military experience while living in barracks with open showers, also high school PE and sports activities, I have showered with a great many both black and white people.
This is something I never really though about, but now that I am, I can’t remember anyone using a washcloth. Simply no place to leave it to dry.
Some people refer to it as a facecloth, normally used for bathing outside of a tub or shower enclosure, and also used for exfoliating purposes.