Do Japan not have child protection laws? How can they just leave someone underage in an apartment and no one seems to care? I know it is what the author wants to set up the story but it bothers me
As I understand it (so take everything that follows with a huge grain of salt), Japan is
extremely hands-off when it comes to family matters that don't directly affect the public; you know all those manga and anime where a side character is getting abused or is being forced to marry or something, the protagonist tries to help, and then gets shut down by being told it's a "family matter" and, thus, none of their concern?
Same thing. Japan (and most Asian countries) is big on familial piety, so if it's another family's affairs, it's considered to be nobody's business but their own. It's why things like hikikomori even exist... Well, that and it's "shameful", so any issues like that (and every other mental health issue) just doesn't get talked about.
This isn't universal, obviously, and manga tends to exaggerate
most things with a real life basis, but a general rule of thumb is that "if it's a family matter, everybody should just mind their own business" applies unless explicitly stated otherwise.