Their relationship would also be different in terms of Naruto's growth as a ninja. Minato could introduce Naruto to different ninja techniques and strategies at an earlier age. They could even develop new jutsu together, which would enhance Naruto's standing in the ninja world and also make their relationship one based on mutual respect for each other's skills and abilities.
The emotional bond would be stronger. Naruto wouldn't have to face the pain of growing up without a father. Minato could be there to cheer him on during his training and be a source of comfort when Naruto faced difficulties, like when he was bullied as a child. He could also help Naruto better control the Nine - Tails' power from the start.
Minato being alive in fanfiction could mean more guidance and protection for Naruto. It could also lead to new alliances and conflicts within the ninja villages.
If Minato and Kushina were alive in Naruto's story, Naruto would have grown up with his parents. He wouldn't have had to endure the loneliness he did as a child. Minato could have trained Naruto from a young age, passing on his powerful jutsu. Kushina could have taught Naruto about the Uzumaki clan's secrets. Also, the village of Konoha would have been a different place, with Minato's leadership and Kushina's strength adding to its security.
Well, in certain fanfictions, their relationship could be more of a slow - burn type. Maybe they start off as just friends or comrades who gradually realize their feelings for each other over time. This would give more depth to their characters and how they interact with each other.
The relationship would be very different. Instead of being an orphan, Naruto would have his parents around for all the important moments in his life. Minato could teach him about being a Hokage from an early age, and Kushina could give him the motherly love and advice. There would be a strong family bond. Naruto would be more confident in his abilities because he has his father, a great ninja, to guide him. And he would have a more stable emotional state with his mother's care.
It could completely change the dynamics of the Naruto universe. Their presence might have prevented many tragedies and shaped Naruto's life differently.
Another aspect could be a time - travel fanfiction. Naruto gets sent back in time and meets a young Minato. They train together, and Naruto shares some of his future knowledge in a veiled way. This could lead to a different development of Minato's skills and a unique bond between them.
In some fanfictions, Minato saves Naruto physically. For example, he might use his teleportation jutsu to appear right in the middle of a fight and whisk Naruto away from danger. In others, he could save Naruto spiritually. He might enter Naruto's mind, like in the fanfic 'Minato's Mind - Saving Mission', and give him the strength and confidence to overcome his inner demons. Sometimes, Minato saves Naruto from a political situation. For instance, in 'Minato and the Hidden Leaf's Politics', he uses his influence and wisdom to protect Naruto from being used as a political pawn.
One interesting fanfic could be where female Naruto, with her father Minato alive, has to face new threats in the ninja world. In this story, she might train under Minato, learning his signature jutsu like the Rasengan in a whole new way. It could explore how having a father figure changes her path from the original Naruto's story.
One idea could be that Naruto grows up with his parents in a world where the Nine - Tailed Fox attack never fully happened. Minato and Kushina could train Naruto from a young age, passing on their unique jutsu and wisdom. Maybe Naruto would be less of an outcast and more of a confident shinobi early on.
It would be very close. Naruto has always longed to know his father, so when he travels back in time to meet Minato, they would quickly bond. They'd probably start training together right away.