The same argument could be made for each time you go to sleep. That the 'you' that's conscious ends to never exist again and the one that wakes up has all the same memories and body but is no longer the same stream of consciousness that went to sleep, not even knowing it's only minutes old and destined to die within hours.
'You' could have effectively lived and died thousands of times in your life and not even be aware of it.
You cannot step into the same river twice - Heraclitus, ~550 BC
We are all a series of continuous evolution, alteration and change. "I" am not the same person who began this sentence. The idea that "I" cease to exist overnight and begin anew in the morning is meaningless. There is no one version of me. I live - and to live is to change!
Oh, wow! It was the ST:TNG episode Second Chances (linked in that article) that got me into thinking about it, and it's really trippy to read that somebody else came up exactly the same thought experiment with a human-replicator, too, and came to the same conclusion that I did: Both the original and the duplicate would have exactly the same memories of entering the replicator, so both would have the same continuous experience of the subjective "I". But if only one existed before replication, where did the second consciousness come from?
After I heard the Radiolab episode, "Loops,", I realized that the only way to resolve the paradox is to figure that our consciousness is re-created more-or-less continuously from our memories. That episode covered the case of a woman who experienced Transient Global Amnesia, which sent her into a loop of about 90 seconds of essentially the same conversation over and over, for hours. There's a famous video of it. That fits with the evidence, from neuroscience, that our consciousness drops out briefly every minute or so while our brains attend to sensory input from the environment.
The COVID-19 pandemic really brought this home to me in a visceral way. In the early weeks, when the CDC was warning about surface contamination, and how I should not touch the mask I had to wear at work under any circumstance, my nose would invariably start to itch. I would tough it out, exercise will power not to scratch the itch, and it would eventually go away. Soon, I realized that I never once got to feel the moment of relief when the itch faded. Always, I would simply notice that it had been gone for some unknown amount of time. It went away with one of those consciousness resets.
So, yeah, like the other folks say, we don't have a continuous conscious experience. The old "I" passes away within seconds, to be replaced by a new "I" with my memories, in a never-ending process of renewal. Think about that next time you walk into another room and forget why you're there.
Personally, I'd say nothing. Or, at least, whatever you say makes you, you. I don't think there's an objective/natural definition for who I am and what is and isn't a part of me. The idea of "me" is kinda made-up, so there's probably no right or wrong answer as to what exactly I label as "me."
I'm probably just saying nonsense, but this is the most coherent answer I got lol
That's a question no one has yet been able to answer definitively though both neuroscientists and philosophers are trying.
I'm of the opinion that "I" am a pattern, encoded in the physical interactions of my brain and body. I'm not certain if I have free will or just like to think I do. But I do believe that whatever makes me "me" is fully contained within the dimensions of my physical being.
The way out of the riddle is that there never was a ship of Theseus to begin with or a you those are just referents like pointers used to refer to an evolving system with a known state at a known starting point and probabilistic predictions of a future state based on known factors.