In Japan same sex marriages are illegal, so to get around that, one will adopt the other. Which makes them officially family (in this case father and son, which may sound iffy but since they're both adults it's not really paid attention to). They weren't adopting the girl (the part where she asked if she could put her name on the paper too but was declined).
Just adding to that that it's a common practice where there is no marriage or some equivalent or partial equivalent of marriage (like a civil union or something) for gay and lesbian couples. Before we had civil unions in Germany gay couples used to do it too, the older guy adopting the younger one. Not only does it make them family but can help in official situations like hospital emergencies (visiting rights and all that), death, insurance, taxes and inheritance.
They trust and love each other enough to to that. I really loved the manga but don't get the family register bit. So are they adopting the child or the three of them are under the same family register. If they are adopting her then that would be so fucked up on the mom's part or if the four of them are under one register that isn't very realistic either so....