I've only worked in the private sector in Japan, and in every company I have worked for (unlike most Japanese people, I have changed jobs a few times), if anyone was sick, they took nenkyu (their annual leave), as no byokyu (sick leave) was offered. In practice, almost nobody took their complete annual leave entitlement, as the working culture of the company pressured them not to.
DW does work in the public sector and says that she can take menstrual leave as sick days, and the organisation is obliged to grant it, but she if she took a sick day, she would then have less holiday to take. She is a rare case of insisting she gets all her annual leave, despite her boss often criticizing her for this and trying to bully her into not doing so. Fortunately, she has an uncommon skill that the company need, so they can't tread on her rights the way workers' rights very often are trodden on in Japan.
Here is the relevant article of the Japanese labor law
Article 68?When a woman for whom work during menstrual periods would be especially difficult has requested leave, the Employer shall not have said woman work on days of said menstrual period
www.japaneselawtranslation.go.jp/law/detail/?yo=%E5%8A%B4%E5%83%8D%E5%9F%BA%E6%BA%96%E6%B3%95&ft=2&re=02&ky=&page=1
To me, that reads as if an employer can't refuse a request for leave, rather than that leave over and above the usual leave entitlement is available. It's open to interpretation I suppose.