I think Shiori is supposed to be the "normal" human of the group.
I'd have to see the kanji or kana used for Koseki, but a Koseki is a family register in Japan, and Bijou is a type of jewel in French. Since she's known as "The Jewel of Emotions" that's probably what they're going for. Koseki could also mean small so she could literally be "Small Jewel".
....her name is The Family Jewels isn't it...