It could be due to inconsistent characterizations. If the author changes how Naruto acts or thinks from one chapter to the next without a valid reason, it gives an impression of instability. For example, one moment Naruto is brave and determined, and the next he's cowardly and indecisive. Also, if the world - building in the fanfiction is shaky. Let's say the rules of chakra or the ninja villages keep changing randomly, it would be considered unstable.
It could mean that the Naruto fanfiction in question has an erratic or unpredictable plot or character development. Maybe the story doesn't follow the normal patterns or logic within the Naruto universe.
In some fanfictions, Naruto might become unstable if he discovers some dark secrets about his past, such as his father's actions having more negative consequences than he thought. It could also be due to the influence of a powerful and malevolent enemy who has the ability to mess with his mind, like using genjutsu continuously to make him doubt his reality and purpose.
In fanfiction, Naruto being emotionally unstable can be a way to explore his character deeper. It could be due to some unaddressed trauma. This can lead to changes in his usual optimistic and determined self. His enemies might take advantage of his instability. It can also cause internal conflict within him, like a struggle between his desire to be Hokage and his emotional turmoil, which could make for a very interesting read.
It adds more depth to the genre. By showing Naruto as mentally unstable, it allows for different types of stories. For example, stories about his recovery or his struggle with his mental state can be created, which makes the fanfiction more diverse.
It could be due to certain events in the fanfiction's unique plot. For example, if Naruto fails a really important mission and blames himself, it could lead to emotional instability. Or perhaps he discovers some dark secrets about his family that shakes his mental state. Writers often use such emotional upheaval to drive the story forward and explore different character arcs.
In some fanfictions, Naruto might be depicted as mentally unstable for various reasons. One could be the extreme stress he endures. He has lost many people in his life, like his parents at a young age. This could potentially lead to psychological trauma that some fanfic authors might explore and exaggerate, making him seem mentally unstable.
One common element could be a distorted view of the characters' personalities. For example, Naruto might be portrayed as more erratic or self - destructive. Another might be a darker, more hopeless world - building compared to the original Naruto universe.
The fem Kyuubi could be initially attracted to Naruto's struggle with the Kyuubi inside him. She decides to help him. As Naruto's mental state improves due to her efforts, they could develop a romantic relationship. For example, she could teach him how to better control the Kyuubi's chakra, and in the process, they fall in love.
I'm not really into promoting fanfics that involve mentally unstable themes as they can often be quite dark and disturbing. However, you can try searching on some major fanfic platforms like FanFiction.net or Archive of Our Own and use the relevant tags to look for such works, but be cautious as they might not be suitable for all readers.