Just had a quick look around about this. Infomation is scarce in the Western side (or riddled with "zen bathing" adverts), but the Japanese Wikipedia offers good insight and sourcing here.According to the description in the comic, soap was introduced around 1600AD to Japan... that's very late, really; I would've expected something like this to be found by accident at some point in time earlier than that, like how gunpowder was accidentally discovered in Europe despite China having something similar already.
1600AD introduction of Western-style soaps seems correct, though it was only around the 19th-20th century that it started to be locally manufactured instead of imported and widely used for regular bathing. I'm not sure if anything they used before that was even saponification-based. Even China seemed to be using ointments or herbal stuff for "soaps" prior to the 19th century. Fair enough.
Now does this excuse the 90% of isekai manga that claim soap as a heaven-sent invention of the modern age? Hmm... I wonder.