First you have to bear in mind that nearly all of the descriptions that you hear about this theory are worded very badly, and are usually based on wildly inaccurate misunderstandings of it. For example, the various universes don't interact at all - there's no possibility of "crossovers" like we heard about in this show, because that's not an aspect of the mathematical definition of the quantum wavefunction.
Everett's interpretation is based on the idea that the universe is defined by its total wavefunction. So all of the different possibilities expressed within that one wavefunction are simply facets of the one true reality that's the sum of all of those possibilities. There aren't "many worlds" or "many universes" - rather, all of those universes are simply facets of the one universe, like the facets of a single diamond. They don't exist - they can't exist - without each other.
So there are infinities of facets of the one universal wavefunction where the part that's "you" and everything that you can perceive, do not diverge....where the component of the universal wavefunction that we call "you" remains exactly the same, and only other parts of the universe diverge (often in vanishingly minute ways). In Everett's interpretation, "you" are defined by your collective wavefunction: the iterations of "you" are just aspects of that one whole, which is in turn an aspect of the one whole universal wavefunction. It's as if your wavefunction is the "real you" and all of the various permutations of your wavefunction are just reflections in a house of mirrors.
It's a totally different way of looking at reality, and one that I don't believe for a second, but in this model your wavefunction is the only "true reality," and the iterations are just facets, shadows, echoes, reflections...however you want to look at it.