Imo, a major aspect of Kiara as a character is that while she projects onto The Little Mermaid, she doesn’t parallel her in a narrative sense, and that due to her inability to truly love others, she fundamentally doesn’t understand the narrative she loves so much.
During her childhood, where she was isolated and bedbound, one of the cultists gave her the book and so it was her only window info humanity and how “real people” behaved. For her, the characters of The Little Mermaid are idealized figures, and real-world humans not matching up to that ideal is what leads her to spiral down into who she becomes in CCC - a woman that doesn’t understand or caste for humanity, seeing herself as the only human.
(I want to point out that according to her bond CE, “on the brink of death, the girl holds a single book she treasures” - it’s possibly her only personal possession, or at the very least the only thing she has ever treasured, which makes her misunderstanding even more tragic!)
So, as a quick summary of The Little Mermaid : a Mermaid (who has a long life but no soul) saves a prince from drowning, and learns from her grandmother that humans have short lives but souls that ascend to heaven when they die - basically, humans have immortal lives after death.
She wishes to obtain a soul, but according to her grandmother,the only way for a mermaid to do so is to have a human love her so much that they share a soul. After this, she makes a pact with the sea witch to go on land - she trades her voice and tail for legs, and if the prince her she will gain a soul, while of the prince marries another woman she will turn to sea foam (as mermaids do then they die). However, the prince is already in love with a girl who saved him after the mermaid took him to shore, though he does care for the Mermaid as a sister.
Eventually, he is about to marry the other girl, and that night the mermaid’s sisters bring her a knife - if she kills the prince she will be able to return to being a mermaid. But seeing the prince and his wife, she can’t bring herself to kill him, and turns to foam. Afterwards, she meets “spirits of the air” that tell her that even if she doesn’t have a soul, if she works for 300 years as an air spirit, she can gain a soul.
It’s history is also incredibly important and interesting - Hans Christian Anderson was someone who was constantly faced with unrequited love, both towards women and men, and was also very occupied with the theme of religious salvation through love. So, him writing a story in which someone, despite their lack of a “soul”, is still able to gain a soul through good deeds as a reward for selfless love is very thematically resonant and also completely understandable.
CCC!Kiara funamentally lacks that sort of selfless love. While FGO!Kiara, still having a bit of who she was before the influence of CCC Kiara can learn selfless love through her relationship with Ritsuka, CCC Kiara is too far gone. She cannot conceptualize selfless love - she could never do what the little mermaid did and sacrifice her life for someone else’s happiness, because for her, other people are fundamentally not human and therefore don’t deserve to be saved, and their joy shouldn’t be prioritized; she can’t understand that the Sakura AI’s are beings that can feel real and legitimate love. No matter how much joy and comfort the book brought her, Kiara was never able to grasp it’s true meaning.
Unlike Kiara, BB parallels the mermaid, being an “inhuman” being with a potentially long life, but no “soul” in a sense. While Nasuverse humans are able to return to the root (the parallel of going to heaven for the Nasuverse), she’s an AI and so she can’t. This is further emphasized by her role in CCC - in the CCC Route Ending (which is the closest thing CCC has to a true ending) has BB sacrifice herself in order to save Hakuno and Sakura. Just like the little mermaid, she gives up her life to save the one she loves (who doesn’t love her back).
Unlike Kiara, BB is capable of the sort of true, selfless love that Andersen clearly admired, which I find to be super fascinating, and I love how FGO acknowledges that dynamic by having the summer mermaid Kiara embody Yaobikuni (who serves as a much closer parallel, being a holy woman that ate the flesh of a mermaid and became immortal) and Shen (a clam monster that creates mirages and changes it’s shape, which is what Kiara does very well!), and not the Little Mermaid herself.