It could be that the wolf is a reincarnation of an ancient protector of the ninja world. It was waiting for the right person to bond with, and Naruto, with his strong will and desire for peace, was the chosen one. So they became partners.
One possible storyline could be that the female wolf partner has special powers. For example, she might be able to communicate with other animals in the Naruto world, which helps Naruto in his missions. Maybe she can sense danger from afar and warns Naruto in time.
The female wolf partner could enhance Naruto's combat skills. She might be really fast and strong, and through sparring with her, Naruto becomes better at fighting. For example, she could teach Naruto some unique wolf - style combat moves.
In the fanfiction, a possible backstory could be that Naruto, after achieving great power and peace in his own time, decided to use his abilities to explore the past. During this exploration, he accidentally or purposefully got involved in the lives of his ancestors. He might have fallen in love with a woman from the Uzumaki clan in the past and had children. These children then passed on their genes through generations until Kushina was born. It could also be that there was some sort of experiment by an ancient shinobi that went wrong or right (depending on how you look at it). This experiment somehow connected Naruto's future self to be Kushina's grandfather by altering the fabric of time and family relations in a very strange way.
Maybe he had some bad experiences with weaklings who let him down in crucial moments. For example, a weak comrade could have failed to protect someone important, and that made Naruto start to dislike those he considered weak.
Maybe Nagato had a relationship with an Uzumaki woman before he became fully involved in the Akatsuki. This woman could have given birth to Naruto but for some reason, Naruto was separated from Nagato at a young age. It could be due to the chaos and danger that Nagato was getting into with his plans for the Akatsuki.
Some possible names could be 'Fang' as it's a simple and common name for a wolf in many stories. It gives the idea of the wolf's fierceness.
There's a fanfic where Naruto's wolf partner is from a hidden clan of wolf spirits. The wolf, let's call it Akela, teaches Naruto ancient wolf techniques that are long forgotten in the ninja world. This makes Naruto even more powerful and gives him an edge in battles.
Perhaps in this fanfiction, Naruto travels back in time. He goes to a time when the world was being formed, and through his unique chakra, he gives birth to the Nine - tails. This could be part of a grand plan to ensure balance in the future ninja world, and he is the only one with the power to do so.
Well, it could be due to a bloodline limit that was previously unknown. Naruto might have some distant ancestor with the ability to transform into a wolf, and through some circumstances, this latent ability surfaces. Or perhaps he made a deal with a powerful wolf - spirit in a mysterious forest, and as part of the deal, he gained the ability to turn into a wolf.
In fanfiction, having Naruto as Itachi's partner can lead to some really interesting scenarios. Naruto's positive and determined nature could contrast with Itachi's more reserved and burdened personality. They might work together to uncover secrets within the ninja world or face powerful enemies. Naruto could also learn a great deal from Itachi's skills and wisdom.