Guys! Nobody has talked to Kon before, besides Furuka. This is just a theory, but I think she might be a ghost or the monster herself. She says everyone in her school ignores her. In her house, none of her family members never addressed her and when her brother says "sis", he could be talking to the other woman who was coocking. It was never stated that she was his mother.
When that bully girl talked about "lazzing about with that thing over there" she could be talking about the abalones instead of Kon herself. This is just a theory, but I re-read the whole thing and it seems the only one who can see Kon is Furuka.