It often starts with a shock or surprise at the mpreg situation. Then, they gradually learn to rely on each other more. For example, Vegeta might start to show a more caring side that he usually hides, and Goku might become more responsible. They'll have to communicate more, sharing their fears and hopes about the future with the baby.
The fanfiction might make their relationship more equal in a different sense. Before, there was always a bit of a power struggle between them. But as mates, they have to respect each other on a whole new level. They would have to consider each other's feelings and opinions more. For example, when making decisions about how to face a new enemy, they would have to consult each other rather than just going with their own instincts like they used to.
The relationship could develop from a simple rivalry. They constantly compete to be the stronger one. However, in a particular fanfiction, during a life - threatening situation, they have to rely on each other completely. This makes them realize that there's more to their relationship than just competition. Female Goku might show concern for Vegeta in a way that surprises him, and he starts to have feelings for her. And from there, their relationship evolves as they explore these new emotions.
In some fanfics, there might be such out - of - character scenarios for the sake of creating drama or exploring different relationship dynamics. But in the original Dragon Ball series, this would never happen as the characters have their established relationships and personalities. Bulma is with Vegeta and Goku has his own family and relationships that are very different.
In these fanfictions, Goku's character might develop by becoming more in tune with his Saiyan pride under Vegeta's influence. Vegeta, on the other hand, could become more open - minded and less rigid. For example, Goku might start to understand the importance of Saiyan traditions more, and Vegeta could learn to relax a bit and enjoy life on Earth more.
In this type of fanfiction, the 'slave' concept is often not literal slavery in a cruel sense. It could be more like one character being in a position of servitude due to a debt or a magical spell. For example, Goku might be 'enslaved' to Vegeta because he lost a bet or made a mistake that requires him to serve Vegeta for a while.
In some fanfictions, it might be a really strange and unique concept where Goku is depicted as Vegeta's pet. But this is far from the original relationship in the official Dragon Ball story. It's just a creative and often humorous take that some fanfic writers explore.
Goku's storyline would be affected in several ways. For one, without Vegeta, Goku might not have the same intense rival that pushed him to become stronger. Also, the Saiyan lore that Vegeta could have passed on to Goku would be lost. This could change Goku's understanding of his heritage and his place in the universe.