I'd argue that a branch of me is still me, in many meaningful ways. This is true for the many-worlds interpretation, where the universe branches, and for multiple simultaneous mes from mechanical copies.

After the copy, my meatself and my electronic self(-ves) will diverge, and will be different entities who only care about each other as others, not as selves. But that's true of cross-temporal and cross-universe entities that have a direct causal relationship as well. I care less about an alternate-world me than about the me I'm currently indexing. I care less about any specific future-me based on how distant it is from my current experiencing.