Isn't Gay Marriage Legalized In Japan Now?
What happened just recently was a court ruling that said the same-sex marriage ban is unconstitutional. Just year before, other court said the ban is constitutional.
That opens the door to discuss the matter in political sphere seriously. But you still need to modify the Constitution to remove the bit "one man and one woman" from it, or find some kind of workaround.
Which is hard, because main party is quite conservative and tries to do all the same delays that were done in many western countries as well. Right now Japan is in the phase where it teeters between allowing "civil partnerships" across the country or not. Many locations have their own rules of that kind already, but there's no countrywide ruling.