I believe it’s more like partnership certificate. Which they acknowledge that they are married but not legally.
There is only one area in Japan I forgot where it is, that gay people can live there and be registered as partners, however if you move out of that area you are not legally considered as partners, that's why most of them adopt each other so they can legally be a family. Also even if you get married outside of Japan they don't recognize that marriage certificate and still consider the partners as strangers. Correct me if I am wrong in any of the info that I provided
Same-sex marriage is not recognized in Japan. As of 1 March 2021, 75 municipalities and three prefectures issue same-sex partnership certificates which provide some benefits but do not offer any legal recognition.
All the comments from up here express the situation in Japan. plus, if you read the manga "until i meet my husband" clearly, you may noticed they were offered adoption suggestion too but they didn't want it because he want to be husbands and infact it took them long enough to get the partnership done because it's newly recognized.
If same sex marriage in japan isn't allowed what about the story about "until my husband arrives*???