Yup, fate. I wanted to talk about this before, but I tend to forget. XD I found that concept to be an extremely important plot point in the entire series, and I only realized it when I reread a part of Moon recently. Fate is a property of Aurora that serves as its memories that it can come back to whenever it wanted.
This was best explained during the mini-Theory that the Occult Club played around in Moon. The horseshoe crab thing that Shizuru came up with caused the collapse of one world, but in the next world they simulated, the mini-Theory just naturally brought a more sustainable species of that horseshoe crab without anyone commanding it to be.
Basically, Aurora can naturally carry memories of the world, and this concept is the basis of creating the Mistletoe familiar and also how the Holy Women passes memories after death. I think this is also why Chibimoth and Kotori’s parents gained memories of their past selves.
This property of Aurora also gravitates people to whom they have heavily interacted in the past worlds. That may as well explain why Kotarou encountered a young Midou and his friends in the war-torn country, and also why the five heroines just flocked together after the war and became Touka’s children, and why they incidentally summoned Kotarou who has become a familiar.
I think this should explain why Kotarou and Kagari gravitated together so strongly that they meet each other in every single world.
Edit: I should have mentioned that this is how Kagari’s Beacon (aka the flaming choices) works. And also Terra ending with the Occult Club meeting the Girl on the Moon. XD