What is the essence of myself that reincarnates?

Let's say I'm Brad, and then I pass away. I then reincarnate again as Tim. Then let's say, in some science fiction way, my past self Brad and Tim meet. Their bodies are different, perhaps their personalities are different, but yet they're the same ''essence''. Now the question is what is that essence? Some people would call it the individual soul.

Let's use another science fiction example. Let's say Tim meets another alternate universe Tim that looks exactly like him. It could be possible this other Tim is another soul that has reincarnated into a body that looks identical to Tim. While Tim and Brad are the same soul chronologically moving from one body to the next, the soul of Tim and alternate universe Tim could be completely different. So what is that essence that goes beyond Tim's body and occupies other bodies through reincarnation?

I know that Atman and Brahman are one. I think I understand the concept that there are no others and there is only Brahman. But in the world of duality, there still ''appears'' to be separation. For example let's say Sally passes away then she passes become Brenda in her next life. Sally and Brenda's ''essence'' goes through a different chronological timeline than Brad and Tim, so clearly they're two different ''souls''...Even if ultimately everything is Brahman. But what is this? Is this Atman who then is indistinct from Brahman? Is this Jiva? Jivatman?