I've never heard 'washings' used the way you suggest... even in detergent commercials, they'll talk about "multiple washes" to refer to an instance of a machine wash.
"Washes" is the plural of the noun "wash". "Washing" is a verb, but you can, in some contexts, refer to an instance of washing as "a washing". So it wouldn't be wrong to say "multiple washings", if you were focusing on counting the actions. But it could still sound weird - imagine if someone said "I've had 5 bathings this week", and not "5 baths". Saying "bathing" sounds like someone else has bathed you.
I've just tried a search of "multiple washings" and I don't really agree with the grammar of the few instances I found. The first result I found is talking about an item having "durability to multiple washings". That preposition "to" just doesn't work with 'durability'. We say something has resistance to things, but 'durability' is a quality. Basically, the whole sentence is rubbish, so I don't trust the use of "multiple washings" either.