It might also be about how Naruto is treated or perceived in the fanfiction. If the other characters see him as a monster, either out of fear of his power or because of some misinformation or plot device, then that would create the feeling. This could lead to interesting storylines where Naruto has to deal with being ostracized or hunted because of this false perception of him being a monster.
Another way could be in the way his powers affect him. Maybe the use of his powers starts to corrupt him physically or mentally. His eyes could turn into some sort of inhuman or demonic - looking form, and his mind could be filled with uncontrollable rage and bloodlust every time he taps into his power, making him seem like a monster.
Maybe it's some kind of special jutsu or a curse that backfires in a way that numbs his pain receptors. For example, a powerful sealing technique might have been placed on him accidentally or on purpose that has this side - effect.
One great example is 'Dracula' by Bram Stoker. There's the iconic vampire, Dracula, who is a powerful and terrifying monster. He has superhuman strength, the ability to turn into a bat, and he preys on the blood of the living. The story is set in spooky locations like Transylvania and an old English manor. There are also many mysterious elements, such as Dracula's control over the weather and his minions. All these factors combine to make it a quintessential monster story.
In fanfiction, Naruto could feel sick due to overusing his chakra. He often pushes himself to the limit in battles and training, and this excessive chakra use might take a toll on his body, making him feel ill.
It would be quite an interesting sight. Naruto's energetic and passionate personality might bring a unique flavor to the song. His strong voice, which we usually hear shouting out jutsu names, could be adapted to singing in a really engaging way.
His parents' death also contributed to his loneliness. Growing up without them, he didn't have that family support and love that most kids had. He had to fend for himself and deal with his feelings of abandonment all alone. And in the fanfiction, perhaps his friends are too busy with their own missions or problems, leaving him feeling isolated.
He might focus on his training. By pouring his energy into getting stronger, he can distract himself from his lonely feelings. Naruto has always been determined, and training gives him a sense of purpose.
One reason could be his status as a jinchuriki. People might be afraid of him because of the Nine - Tails inside him, so he feels unloved. Another reason is that he was an orphan, and without his parents around, he may have felt like he didn't have that family love that others had.
In many fanfictions, Naruto might start by training harder. He throws himself into his ninja training as a way to distract himself from the loneliness. It gives him a sense of purpose and also helps him become stronger.