Well, in such a fanfiction, it could lead to Naruto being raised in a different environment. Maybe he would be found by other shinobi who would train him in a different way. This could result in Naruto having different skills and abilities. His view of the world might also be less about proving himself to the village that rejected him as an orphan. Instead, he might be more focused on other things like exploring the ninja world outside of Konoha.
If Kushina and Minato abandoned Naruto in fanfiction, it would completely change the Naruto's character development. Naruto grew up lonely and seeking recognition because he was an orphan. Without their abandonment, he might have had a very different personality, perhaps more confident from the start. Also, the relationships in the village would be different. For example, Iruka might not have had the same special bond with Naruto as he did in the original story.
It would be a huge shift. Naruto's motivation in the original story was so much about his lack of parents and the desire to be acknowledged. If Kushina and Minato didn't abandon him, his main drive could be something else entirely. The Akatsuki might not even see him as a target in the same way, since his situation would be different. And the whole dynamic between him and characters like Sasuke, who also had a troubled family background, would be altered.