All right!
Sooo, story time:
Way back before Vivi was born, her father used to be completely different. He was lazy, selfish, loud and rude. Back then, he, his wife and their newborn son lived in what was essentially a shack that was falling apart in pretty much the middle of nowhere. He wouldn't work or take care of his kid and constantly be out of the house. And when he wasn't, he'd just sit on the couch, drink and watch tv while complaining to his wife about every little thing bothering him. One of these days he went to a bar, got a bit too drunk and fooled around with someone. This act ended up creating Vivi, something he'd have to confess to his wife when they found an egg left at their door addressed to him.
Understandably furious, Diana whipped her husband into shape and gave him one last chance along with an ultimatum. She had enough and if it wasn't for the fact that they had not only one but now two children, she wouldn't have put up with all this. The espeon was past the point of caring for her marriage, but didn't want these innocent children to have to grow up with separated parents, so she endured way past the point where others were telling her to stop and leave. This was a sort of awakening for Sebastian, finally realizing what he was doing and that he didn't even deserve this last chance that has yet been graciously given to him. From that moment on he vowed to change for the better, not just for himself but for his family. He wanted to give his children the best life possible and worked hard alongside his wife to get them out of their current situation and into a better home.
Their first goal was to move out of this barely livable home, but they both needed to save up for that. Diana felt awful neglecting her children for the sake of work, but she agreed with her husband that this was no place to raise children and that they needed to leave asap. So they worked together to save enough money to move out of the old, broken shed and into a proper home.
However, because both his parents were busy all the time, Kai quickly had to learn to take care of himself. And soon after he'd also feel like he had to take care of his little sister once she hatched. At least whenever their parents weren't around. Due to this, he built a strong bond with her as he was essentially acting as a third parent for her. He cared deeply for the other eevee and was willing to do whatever it takes to make her happy. Something he'd quickly prove. During a particularly cold and strong winter, the building they were staying in could no longer withstand the freezing temperatures and equally cold winds, especially with the lack of functioning heating appliances and falling apart walls. Kai couldn't bear watching his sister freeze and shiver all day every day, and there was only so much warmth huddling together under a blanket could provide. However, he did learn of the various evolutions that his species could turn into through a children's book, and he also understood that it took a special kind of stone for it. He learned of flareons and the warmth they can provide and figured that if he became one, neither of them would have to freeze anymore. Now, normally since this is an irreversible, life changing decision, parents tend to wait until adulthood before they offer their children the opportunity of evolution. However, he begged his parents for a stone so he could provide warmth to his sister. At first they rejected him since he was still way too young to make such a decision. However, Sebastian, ridden by guilt of his past behavior and neglect, as well as a desire to make things right, bought a fire stone with the family savings behind his wife's back and gave it to Kai in the name of helping Vivi. Understandably, Diana was furious that he made this decision for him at such an early age, but after seeing how happy Kai was and how Vivi was no longer shivering and whining all the time, she let it slide. Probably because she too was feeling immense guilt for having her children be freezing all the time. Kai didn't seem to show any regret for this decision, not even to this day, something that over time eased Diana and Sebastian's conscience.

Fast forward some years and things have changed quite a bit.
No longer did they live in this run down shack. Instead their father somehow managed to land a well paying job at a major company, and thanks to it they were able to afford a nice house in the outskirts of a big city.
Diana and Sebastian’s marriage has mostly been revitalized after the jolteon has proven to have changed for the better and showed his dedication to it over the years. He was essentially a completely new pokemon, and much more like the one Diana fell in love with all those years ago. This did not undo his past however, and he is still scared of it to this day. Scared that his children would end up becoming the same way, lacking discipline, growing lazy and eventually hurting those they loved. Out of fear, he became strict and overprotective. "No going out past 6PM, no going to the beach without a parent, no staying up past 10, and Vivi if you try and reach for the cookie jar again you're grounded for the rest of the week." That along with his unfiltered complaining for any kind of flaw in his kids made him fairly unpopular among them. "Kai, your grades have been going down, you're slacking. Vivi, all those sweets you keep eating are making you fat, you're already getting chubby."
He knew he wasn't being "nice" but he was doing it all to prevent the same things he did from happening again. It was his own way of showing love, by trying to make them become the best selves they can be with the brightest future possible.
Luckily for them though, Diana was there to sort of even it all out. Unlike her husband she was soft and understanding and even kind of encouraged them to not listen to him and live their lives however they wanted (within reason of course). She'd calm down tense situations when he became angry and make him ease up on his rules, stand up for them when they did break them and be a loving and approachable parent when needed. Not that Sebastian couldn't have moments of love, kindness and softness himself, but those were far between. Usually if Kai or Vivi needed someone they could trust and turn to, it was their mother, making her the popular one among the parents. Together the parents created a good dynamic of making up for what the other lacked. Sebastian brought order, direction and rules to help guide them, while Diana offered protection, trust and freedom when things got too much.
Despite Sebastian's behavior, Vivi still gets along well with him. She learned to kind of roll with the punches and not take them that seriously, seeing as her mother was encouraging her to do the same. His rules and lectures became more of an eyeroll that could be safely ignored for the most part. And as she grew older and learned of what he was like in the past, she started to understand him more and forgave him for his behavior. She didn’t like or approve of it, but she saw and understood where he was coming from. That much can not be said for Kai though, as he and Sebastian tend to not really get along that well. They can tolerate another, but Kai doesn't mind barking back when his father gets on his nerves, which happens quite easily.

That is pretty much the family's dynamic and how and why it came to be!
There is of course the mystery of Vivi’s actual mother and that unknown twin of hers buuuut that’s for another time. For now, the siblings are unaware of their existance.

As a final note on the dynamic: Unbeknownst to him, Sebastian has had a major, special kind of lasting impact on his daughter.
You see, one night when she couldn't sleep she came to check in on her parents to see if they could help her. But what she saw that day is something she could never forget...The roughness, the aggression, the intense domination he was showing...The way he snarled, bit and showed no restraints or remorse...Those words he growled she didn't even understand at the time...The way her mother looked pleased despite the things he was doing to her...it was something that stuck with her and a moment she's been chasing to experience first hand ever since. Yet none came close to replicating what she wasn't supposed to see that night...This too most likely affects her relationship with him, whether she’s aware of it herself or not. It certainly shaped her preferences in bed~