In a fanfiction like this, Fem Kyuubi's love for Naruto could be a deep, almost possessive kind of love. Since Naruto has always been different from others in how he treats the Kyuubi. As for hating Kushina, it could be that Fem Kyuubi blames Kushina for separating her from the world for so long. And this could lead to a lot of internal conflict within Naruto, who loves his mother but also has a bond with Fem Kyuubi.
Maybe she was forced to for some dark reason. Perhaps there was a threat to Naruto's life if she stayed, and she thought leaving was the only way to protect him.
Well, in some fanfictions, the idea of Naruto hating Kushina might be a very dark and twisted take. Maybe the author wants to explore extreme emotional scenarios. As for Naruto loving in this context, it could be about different kinds of love - like platonic love for his friends or a romantic love for a character. Fanfiction often takes liberties with the original story's relationships.
Maybe in this fanfiction, there was some sort of dark magic or mind - control involved. Kushina could have been unknowingly influenced to do something that deeply hurt Naruto. Like she was made to abandon Naruto at a crucial moment in his life, which then led to his hatred.
One possible reason could be a misinterpretation of Kushina's actions. Maybe some events were presented in a way that made Naruto think she was neglectful or had made bad decisions in the past.
It might be due to jealousy. If Minato was getting more attention or praise in the village for his achievements while Kushina felt overlooked, that could create resentment. Say, Minato was constantly being hailed as a hero for his speed and battle tactics, and Kushina, who also had great power, felt like she was in his shadow.
The fanfiction could develop this hatred through the introduction of new characters. These characters might feed Naruto false information about Minato and Kushina, making him question their motives and ultimately hate them. For example, a character who has a grudge against the Uzumaki family could manipulate Naruto.
The original Naruto story builds on the idea of family bonds being strong and unbreakable. But in the 'naruto hates kushina' fanfiction, this fundamental aspect is turned upside down. This changes the whole dynamic of Naruto's character development. He is no longer the Naruto we know who is inspired by his parents' legacy, but rather a Naruto with a completely different set of emotions towards his mother.
One possible reason could be a misunderstanding. Maybe some false information was spread about Kushina that made Naruto believe she was a bad person. For example, if there were rumors that she abandoned him for some untrue reason, it might lead to his hatred.
This fanfiction could also change the relationships with other characters. For example, if Kushina hates Minato, the Third Hokage might have to step in more often to mediate between them. And the other villagers might also take sides, creating a divided village atmosphere that is quite different from the united one in the original story.