I am sorry that the material available in Chinese is unlikely to clarify any questions on this point. There are several terms also used for sanitary items or clothing, like 病衣，洗病衣, sme gab, and raja’sco.da. I am using my phone to navigate betwen sutta central and cbeta so my notes are a bit lame.
Msv: Thullananda goes out without a period cloth (病衣)and brahmans see the stain: Two offences, not using and not returning.
Sarv: nuns should return the period cloth (洗病衣).
Dharmaguptaka: not in patimokkha. Khandakas only, nuns may use furniture if they are able to sufficiently protect the furniture (?).
Mahasanghika: nuns should use the anicolaka and not wear it too deep or too loosely. Another rule about not washing rags in public places.
The Tibetan commentary discusses this and introduces a concept of guptibhaga, which my Victorian sensibilities preclude me from translating.
Anicolaka is ambiguous but I would have guessed “ani” (=thigh, body parts above knees) is a euphemism.
Ute Husken (2001) Pure or Clean in “Traditional South Asian Medicine” seemed to suggest that the avasatha civara had a symbolic value to pacify the anxieties of furniture owning hosts. If we look at things that can become impure by contact with menstruating women in say, Leviticus, beds and furniture are one of them. But what stands out from the khandaka account to me is actually that an avasatha civara was insufficient to protect the furniture. So this is not the real purpose of this cloth, which I understand to be about personal cleanliness.
In an Islamic hadith, Aisha is shown as wearing a sarong (izar) during her menses. In some hadith she is shown just cleaning the spots off her normal clothing (at the time she only had one robe), other times (better times?) she has a change of clothing for her period.
The general cultural expectations around menstruation are widespread in India and the Middle east and normally involve bathing one’s person and sometimes clothing after bleeding. I guess the practical reason for having a dedicated sarong for periods would be that it should be washed after “day 4” (if you are lucky lol), which would mean you could keep your normal lower robe as the clean one to get changed into after post-period bathing. But this seems to be a nicety rather than a strict necessity, hence the robe being shared. You would still need to use rags underneath it.
Despite Sanskrit terms like malavadvasas (wearing menstrual clothes) and udvasas (wearing fresh clothes after menstruation), there is not actually a lot of references directly on this point in Sanskrit. I think it is unlikely that nuns ever had a lived culture of shared menstrual rags.