It could be an original character created by the fanfiction author. This character might be a descendant of the original Kyuubi, but with a female form and her own distinct personality.
Well, in such a fanfiction, it might be that Naruto becomes extremely powerful like a god. Since he is trained by the Kyuubi, he could gain unique powers and abilities. Maybe he masters the Kyuubi's chakra in ways no one else has. He could use this power to defeat powerful enemies, protect his village in extraordinary ways, and perhaps change the entire shinobi world with his new - found god - like strength.
Yes, there are likely fanfictions with such a concept. The idea of a 'godlike' Naruto trained by the female Kyuubi is an interesting and popular concept in the Naruto fanfiction community. It allows for unique storylines and character developments.
Naruto might become more in tune with the Kyuubi's power. He could access more of its chakra easily and use it more effectively in battle. For Hinata, she may gain a new sense of boldness. The training could make her less shy and more assertive in combat situations.
The main character is Naruto without a doubt. He is the one on the path to becoming god - like through the Kyuubi's training. The Kyuubi is another key character. It has a vast amount of knowledge and power, and it imparts this to Naruto. Additionally, characters like Jiraiya could be important. Since Jiraiya was Naruto's mentor in the original story, he might play a role in this fanfiction too, perhaps guiding Naruto in a different way in combination with the Kyuubi's training. And then there are Naruto's friends. Sakura could be there to support Naruto or be in awe of his new powers. Sasuke might be an interesting character as he could be Naruto's rival, either trying to catch up to his power or oppose him if their paths diverge due to Naruto's god - like transformation.
The training by the fem Kyuubi could also change Naruto's view of the world. He might start to see the tailed beasts in a new light, not just as powerful and dangerous entities but as beings with their own personalities and stories. This could potentially lead to him trying to bridge the gap between humans and tailed beasts in the fanfiction world.
Well, in this fanfiction, perhaps the female Kyuubi has a special connection with Naruto. She might train him not just in physical combat but also in mental strength. She could expose Naruto to illusions that are based on her own experiences as a powerful being. Through this, Naruto learns to break free from powerful mental restraints and becomes even more confident in battle.
In this fanfiction, Naruto and Hinata might gain unique powers. Since they are trained by the Kyuubi, they could potentially learn to control and utilize the Kyuubi's chakra in new ways. Naruto already has a connection with the Kyuubi, so this training could enhance that bond further. Hinata, on the other hand, might develop more confidence and stronger combat skills, making her a more formidable ninja.
One possible plot could be that Naruto starts off as his normal self but as he is trained by the fem Kyuubi, he gradually gains godlike powers. He then uses these powers to protect Konoha from a new, powerful threat.
In this fanfiction, a major plot point might be Naruto facing new and stronger enemies. With the training from the female Kyuubi, he becomes a target. Enemies could be after the secrets of his newfound power. Also, the training could involve Naruto learning to control the Kyuubi's power without losing himself. This is crucial as in the original series, Naruto often struggled with the Kyuubi's influence. The female Kyuubi might have a different approach to teaching Naruto, which could lead to interesting character development for Naruto as he overcomes his inner demons and gains more confidence in his abilities.