You argue that there is little religious ground to reject homosexuality in Japan and indeed there isn't. However, religion is not the only reason to dismiss gay equality. Cultural effects can also influence the stance of a society towards this issue.
Traditionally, homosexuality was known and tolerated in Japan. However, during the Meiji era the Japanese society developed an animosity towards homosexual practices. Among others, sexology, a pseudo-science, was used as a reasoning against it.
Today, in modern Japan, homosexuality is legal, but same-sex unions are not recognized unless the marriage was in a country where same-sex marriage is legal, nor can a same-sex couple adopt legally. Japanese gay men and lesbian women often conceal their sexuality. This is actually very similar to the situation of many other western countries now or, in those cases where it changed only recently, a few years ago.
All in all I would say that LGBT rights in Japan are not yet progressed as far as in some other countries not because of religious reasons but because of cultural factors. If the development in other countries is any indicator, then a higher awareness of these issues, for example by public outings of prominent members of society, will facilitate the introduction of more LGBT rights.
Sources: Homosexuality in Japan, LGBT rights in Japan