One major implication would be his power level. Tsunade is known for her superhuman strength and Jiraiya was a very powerful ninja. Naruto might inherit a combination of their abilities, making him even more formidable. His relationship with other characters would also change. For example, his interactions with Kakashi would be different as Kakashi would be dealing with the son of two of his former comrades. And his standing in the village would be different, perhaps more of a prodigy - like figure from the start.
If Naruto was Tsunade and Jiraiya's son in fanfiction, his story would be radically different. His access to knowledge and techniques would be much greater from the start. Tsunade could teach him medical ninjutsu at an early age, and Jiraiya could pass on his toad - related jutsu and sage mode knowledge more quickly. His relationship with the Sannin as a whole would be different. Orochimaru might view him as a more direct threat or a more valuable asset. And his place in the village's future plans would be much more defined and perhaps more pressured, as he would be seen as the heir to two of Konoha's greatest.
In the official Naruto story, Naruto is the son of Minato and Kushina. However, in the realm of fanfiction, it's possible for such a concept to exist. Fanfiction writers often come up with creative and alternative storylines. But this goes against the original Naruto lore. It would be a very different take on the Naruto universe, with implications for Naruto's character development, his relationships with other characters like Sasuke and Sakura, and the overall plot. For example, his training might be influenced by Tsunade and Jiraiya in a more family - oriented way rather than just as mentors.
The story could be very different. Naruto might grow up with more access to powerful jutsu. Since Tsunade is strong in medical ninjutsu and physical strength, and Jiraiya is a master of many powerful ninjutsu, Naruto could be a super - powerful ninja from the start.
No, in the official Naruto series Naruto is not related to Tsunade and Jiraiya in that way. But in fanfiction, writers can come up with all kinds of wild ideas. They might write a story where Tsunade and Jiraiya had a relationship that resulted in Naruto. This could lead to interesting plot twists, like Naruto having different powers or a different personality. Maybe he would inherit Tsunade's super strength and Jiraiya's toad - related jutsu. But remember, this is all just fanfiction and not part of the original Naruto plot.
In this fanfiction, their relationship would be transformed. They would be like a married couple in a way, but without the formalities. Jiraiya would probably tone down his womanizing ways as he has a family to think about now. Tsunade would be more open and less prickly around Jiraiya. They would bicker less and cooperate more. Their relationship would be centered around Naruto, and as he grows, they would grow together as a family unit, with their bond getting stronger over time.
One implication could be a change in Naruto's social status within the village. He might be treated differently, perhaps with more respect due to Tsunade's position. Also, his training could have been different, as Tsunade might have imparted her own skills and knowledge to him earlier.
He could also try to talk to them directly, despite the betrayal. Naruto has always believed in the power of communication and the bonds between people. He'd want to understand why they thought he was worthy of betrayal and try to convince them that they were wrong.
In such a fanfiction, Naruto might receive more focused training earlier. Jiraiya could pass on his unique ninjutsu skills more intensively, and Tsunade might teach Naruto about medical ninjutsu. Also, Naruto would likely grow up in a more stable family - like environment, with the support and love from both Jiraiya and Tsunade. This could lead to him being more confident and less of an outcast in the village.
He would probably also feel angry. After all, he has sacrificed so much for the village and for them. He might start to question his own beliefs and values. But Naruto being Naruto, he would not give up easily. He would try to prove his innocence and show them that they are wrong. He could go on a journey to find evidence or seek help from other allies who still believe in him, like Shikamaru or Kakashi.
One possible scenario in this fanfiction could be that Naruto grows up with a more balanced upbringing. Tsunade might teach him medical ninjutsu and how to control his emotions better. Jiraiya, on the other hand, would surely train him in powerful combat techniques and in the ways of the sage. Naruto would probably be more mature and well - rounded at an earlier age, with a combination of Tsunade's tough love and Jiraiya's adventurous spirit guiding him.
He would probably be an even more powerful ninja. Jiraiya could teach him amazing jutsu and Tsunade could train him in physical combat and medical ninjutsu. He'd likely have a more well - rounded skill set.