It’s not entirely destiny, really the best term for this is the red string of fate. This doesn’t have to be necessarily shippy, so let’s look at it from a non-ship standpoint. The Priest and Atem were close- close enough for Atem’s trust to be deeply betrayed to a point of grief and hurt when Seto turned unwillingly against him. In the end of things, Atem left Seto a wartorn country, complicated directions (not to mention the guy’s love interest was dead) and the crown.
There was most likely a lot of regret from Atem’s point of view. Was this ideal? No. Of course not. Seto was a friend, he was a Court member. There was no way he would have done this if it were not completely necessary. The remainder of the Priest’s life would be rebuilding Egypt and trying to get the country to live down Atem’s sacrifice and death. Would he ever be as good a Pharaoh, even with Atem’s short reign? Would he ever fix Egypt? In the end I think, despite the friendship and the bond, he probably had a lot of hostility towards the situation. His life was ruined from that moment on. (And the fact that this is what his father wanted for him in the first place shouldn’t go unnoticed)
So Atem lays in wait three thousand years and emerges a spirit with no memories. Seto’s soul goes on to be reincarnated. The cycle begins again. It might be fair to say Seto’s life is always destined to be screwed, not that he’d believe such a thing. But the pull between friends, the red string of fate, can’t be overlooked. They are pulled towards each other constantly. Kaiba sees the spirit as someone to conquer, he’s an obstacle. The spirit sees Kaiba as many things but most of all Kaiba is a constant. He might not like the way Kaiba operates or conducts his business most of the time but he respects the guy more than almost anyone else.
Kaiba isn’t just some regular asshole that needs to be taught a lesson (yes though he is an asshole that needs to learn many things, so thinks the spirit). Atem isn’t just some guy who gets lucky in his duels. There is mutual respect, there is a shared bond. They are connected in a way that neither question beyond the dueling. It’s just a thing that they have. That they look forward to. The spirit looks for Kaiba on the next horizon- even if Kaiba isn’t his goal, Kaiba is deserving of his attention. And Kaiba lies in wait constantly for the time to be right to challenge the spirit. They both thrive the best in duels when it’s with each other or on the same team.
The way they left things and their relationship in Egypt spawned what became a rivalry in modern day. The red string of fate connects all things, but I think Atem and Kaiba are connected a bit more strongly than most.