So here's my theory on how cossette "got" the spirits before kiara:
I think fake has the power/spell to call for spirits or make them visible for other people because in the earlier chapter it seems that only those who control the spirits manifests them when used. That's why she showed Kiara...
Watch the dad have like a bs excuse the same claude had. "She's the spitting image of her mother, it's too painful to be with her" blah blah blah. Stupid excuse. As a father you should be cherishing your daughter because she's a piece of your beloved that she left behind. It's such a cop-out...