Even more like Ikoku Nikki.
That's true. Basically the same story.
Do Japanese families really just divvy up orphans at funerals like giving away the deceased’s cats to whoever is willing to buy a new litter box?
You gotta see the flipside; familial obligation is still seen as important. So although we tend to see the shitty examples here, in the West the kids would have likely ended up in the dreaded "system". While in Japan at least the family in general takes care.
Although I can't say how the process actually works. But adoption laws in Japan seems to be weirdly lax (see that adult adoption stuff, although I guess that doesn't say anything about how this type of adoption here works).