I don't have a lot of textual evidence, but personally I think No. 13 is more or less a robot copy of Ueno. The main way in which they differ is that No. 13's body is a repurposed caretaker model robot, which explains why she's able to do things like launder clothes in her hands in ch 19, or why she would elect to mow the school grounds in ch 48. As for the "maiden's spirit" element, I can only recall that particular term appearing one other time in chapter 38, when Yamashita is consoling Ueno at the end. I believe this refers to how No. 13 was programmed to have the same feelings and attitude as Ueno, but the regulator/mechanism/whatever goes out of control, meaning No. 13 is essentially always acting like Ueno would in her most emotional/excited state.
Oh, and I think No. 13 doesn't recognize Minamine because they've never met before; No. 13 didn't appear in chapter 37.
That's my read of most of it, anyway. There haven't been character profiles for this manga, so it's a bit vague at best.