Your comment contains an excellent point that can stand independently of Many Worlds (branching all the time). Namely, memory explains anticipation. Anticipation feels like a pointer to the next moment, but it's just an inference based on a long sequence of memories.

There is nothing wrong with anticipating a future experience, but there is also no constraint against anticipating other future experiences as well. And most of us often do; we call that "empathy". We have much more reliable history of knowing how correct/incorrect our anticipations about our own body were, though, and much less ability to ignore those outcomes. So self-concern feels different. But in some sense, self-concern is empathy for our future selves.