Just a bunch of photos I’ve found online… the design of the mugs are interesting, with a “side pocket” to hold soap and other gear. Some seem to have a soap cup that fitted in the top as well.
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Mid 1800’s shave mugs
Wegian Wetshaving
That perfect moment caught between lather and blade; the little breathing space where it's you, your razor and perfection
Lather warming cup, I think, would be the use of the top-fitting insert. Very interesting… online anthropology at its finest!