I've seen the term in historical contexts a number of times through the years, but haven't found a good definition. So I asked a friend* who is extremely knowledgeable about this period of fashion history in particular.
"Washing silk" is simply a silk that can be wet cleaned. Probably the equivalent of a hand wash now. Regular laundry practices of the day were pretty rough, wash tubs/boards and wringers/manglers. Something delicate and pretty like a peignoir wouldn't have been subjected to that kind of treatment, but it would still need more real washing.
The distinction was important because many silks water spot and are otherwise never the same when they get wet. Taffeta in particular gets part of its crisp hand from how it's woven, but there's also residual chemicals left on it from the silkworm cocoons that significantly contribute to this and to its sheen. If it is washed, the hand and sheen permanently change. It still has a little body to it - it's not limp and clingy - but it will never be crisp again and it doesn't have the same appearance.
Sometimes this is done deliberately to change that hand. That is called "washed silk" and it can be done to get just the right flow in the fabric. Or it is approximated by actually sanding the surface so it's a little faded and soft instead of lustrous.
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