Adoption does exist in Japan but it's not as common. It's actually primarily used for families to adopt adult men, in order to keep the family name in case they don't have a male heir to inherit the name. If the man were to marry into the family (by marrying their daughter, for example) the family name would be "lost".
Then secondly, because of the way the family register works in Japan, when a man and a woman get married, the woman is written into the register of the man. In order for someone to adopt a child, they first need to be married, and the family register system doesn't allow for same-sex marriages.