There could also be a shift in the power balance within the ninja world. Other villages might see Konoha as weakened due to Minato's self - sealing and might take advantage of the situation, leading to more conflicts and power struggles among the villages.
One consequence could be that Naruto has a constant struggle to keep Kaguya's power in check. He might experience internal battles within his mind as Kaguya tries to break free.
He might become withdrawn. Instead of being the energetic and outgoing Naruto we know, he could isolate himself from his friends and the village. This would change the whole dynamic of the relationships in the fanfic.
One possible reason could be to protect the village. Minato is known for his self - sacrificing nature. By sealing himself, he might be containing a great evil or power that could otherwise destroy Konoha. For example, if there was a powerful enemy or a dangerous jutsu gone wrong, sealing himself could be a last - ditch effort to keep everyone safe.
One consequence could be that Naruto becomes an outcast in a different way. In the original, he was an outcast because of the Nine - Tails, but here, it could be because of his abandonment. He might not have the same dreams of becoming Hokage. Instead, he could be more focused on survival.
On a more personal level, they could face internal guilt. As Naruto grows up and achieves great things on his own despite their neglect, Minato and Kushina may realize what they've missed out on. This realization could cause them a great deal of internal turmoil and regret for not being there for their son when he needed them.
The village of Konoha would also be affected. With Minato still alive, he could continue to lead the village and make different decisions. This might change the relationships between Konoha and other villages. For example, there could be more peaceful alliances as Minato is known for his diplomatic skills. Also, Kushina being alive would mean that the Uzumaki clan might have a stronger presence in the village, which could lead to new jutsu developments and a different power balance within Konoha.
One reason could be for protection. Maybe in the fanfiction, there are powerful enemies targeting Naruto's chakra specifically. Minato, as his father, seals it away to keep Naruto safe from those who would misuse his power.
Perhaps Minato has been corrupted by power. In the fanfiction world, he might have come across some source of power that has twisted his mind. By sealing Naruto's chakra, he could be trying to take that power for himself. It could also be related to some prophecy that he misinterprets. He thinks sealing Naruto's chakra is the key to saving the world, but in reality, it's the opposite.
It can strengthen Naruto's relationship with his family on a spiritual level. Even though they are sealed within him, he might feel a closer bond to his parents. This could make him more confident in his identity as the son of Minato and Kushina.
Naruto would likely be extremely confused at first. He has always idolized his father, and to have Minato do such a thing would shatter his perception. He'd probably try to talk to Minato, ask him why he's doing it.