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.
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.
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.
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.
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.
Minato would likely be extremely shocked. Naruto is his son, and the idea of Naruto attacking him would be completely unexpected. He would probably try to talk to Naruto first, to find out what is wrong and why he is being attacked.
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.
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.
The relationship can be developed by having Minato appear in Naruto's memories more vividly. For example, Naruto could start having flashbacks of Minato's training sessions or family moments that he didn't know about before. This would make Naruto feel closer to his father.
Minato's presence would likely make them closer. He'd encourage them to see each other as brothers. Since Minato is Naruto's father, Naruto would have more confidence and be more at ease, which would make Sasuke respect him more. They'd have a common figure to look up to and follow his teachings together.
At first, Naruto might be in awe of his parents, especially Minato who is a legend. But as time goes on, they would become more like a normal family. They would have arguments over small things like any family does. However, their love for each other would always prevail. Kushina would be the emotional anchor of the family, while Minato would be the strategic thinker. Naruto would grow up with a strong sense of family values, and they would all be there for each other through thick and thin, which is very different from Naruto's lonely upbringing in the original story.