I'm pretty sure the author just retconned Minami's circumstances on the fly actually; the early chapters really don't jive with what we learn from her employer and see of Shizuku. Just to point out the obvious, I don't know what kind of criteria Japan has for potential foster families of abused children but I'm tolerably certain violently abusive paint sniffers don't make even the long list so either Minami's lying through her teeth back there (presumably covering for Shizuku, who the later chapters seem to imply she's living with) or the whole setup got rejiggered.
There's not any reason to assume any of that. First of all, I don't know much about the japanese foster care system either,b ut I know that in the US system, assholes slip through the cracks ALL THE TIME. Even well-meaning systems have a hard time keeping track of abusive foster parents, because the number of foster kids is usually way bigger than the number of applicant foster homes.
I don't see how you're saying they don't jive. She says her parent hospitalized her because they're a substance abuser, and her boss says her parents (foster or bio) are violent substance abusers who tried to kill her regularly. Now he might have been hyperbolic about that, but we know at the very least they menaced her with a knife.
We don't know if any of the parents mentioned are fosters, or if they're just talking about her biological parents the whole time. It could be that she was taken from her bio parents but they still track her down sometimes. It could be that she was taken in by foster parents who were abusive. It could be that she "lucked" into abusive fosters after being taken from abusive bio parents.
There's nothing inconsistant about any of that.
There is, however, a lot of inconsistancy between your idea of Shizuku and the way she's presented to us. The comic goes out of it's way to show us that their relationship was casually violent and that they didn't really think much about it. We see Shizuku hit Minami twice, but she never does any lasting damage to her, even when she's directly confronted by defiance. She gives up almost immediately without anger or any kind of fight once Minami says she's going back to Iori. That isn't typical for the kind of abuser that would hospitalize someone over a one-night stand. And what reason would Minami have to lie to Iori about who beat her up, anyway? Iori doesn't know Shizuku, or anything about Minami's situation.