Sure. The characters are a huge part of Naruto Shippuden fanfiction. Fanfic writers use the characters to attract readers who are fans of those characters. They might focus on a character's backstory that wasn't fully explored in the anime and expand on it. Also, the interactions between characters in the original show give fanfic writers ideas for different scenarios in their stories.
By their unique abilities. Naruto's Rasengan, Sasuke's Sharingan, these abilities can be the starting point for many fanfiction ideas. Writers can think about how these abilities could be used in different battles or situations not shown in the original.
One of the most popular is 'Naruto Shippuden: Sakura Hiden - Thoughts of Love, Riding Upon a Spring Breeze'. It's popular because it shows Sakura in a new light, with her own adventures and growth.
I don't think it's strange at all. Naruto is a very curious person. Fanfiction can offer new perspectives on his own life and the lives of his friends. He might learn something new about himself or get inspired by the creative ideas in the fanfiction. It could also be a form of entertainment for him during his downtime.
A Naruto x Hinata fanfiction could focus on a mission they go on together during Naruto Shippuden. They are forced to rely on each other in a dangerous situation. Hinata gets injured protecting Naruto, and Naruto is filled with a new - found determination to protect her. This experience deepens their bond, and Naruto starts to see Hinata in a whole new light, leading to the start of their romantic relationship.
Obito's influence would likely make Naruto more aware of the darker side of the ninja world. Naruto is usually very optimistic, but with Obito's guidance, he would see the hardships and betrayals that can happen. For example, Naruto might become more cautious in his trust of others.
One possible fanfiction could be about Naruto and Obito teaming up to face a new, powerful enemy. They could combine their unique skills, like Naruto's Rasengan and Obito's Kamui. Their relationship could start off rocky due to their past differences but gradually turn into a strong bond as they fight side by side.
One great Naruto x Hinata fanfic in Naruto Shippuden is 'A New Beginning'. It focuses on their relationship development after Naruto returns from his training. The story beautifully depicts how Hinata's feelings for Naruto become more obvious and how Naruto starts to see Hinata in a new light.
In many Naruto Shippuden fanfictions where Naruto gets sick, it often shows a different side of his character. Usually, his friends like Sakura and Kakashi would be very worried. Sakura might use her medical skills to take care of him. And his enemies might take this opportunity to try and attack. But Naruto's strong will would still be there even when he's sick. He might not be able to fight as powerfully, but his determination to protect the village would remain unshaken. This kind of fanfiction can explore the relationships in the Naruto universe more deeply when the main character is in a vulnerable state.
One great 'Naruto Shippuden Naruto and Sakura Fanfiction' is 'The Bonds We Share'. It delves deep into their relationship, showing how they grow closer during the Shippuden era. It has some really touching moments between Naruto and Sakura that stay true to their characters as seen in the anime.