You've just described the Ship of Theseus, and there really isn't a right or wrong answer.
Yes the teleported person isn't you, but also no it totally is you.
It's no different conceptually that all cells in your body being totally different from the cells you had several years ago. Yet despite not having those same cells you are you, but also not you. You live in the Star Trek transporter dilemma/Ship of Theseus experiment daily (or atleast one half of it). Infact I'd argue that since the transporters make an exact recreation, it's more "you" that your body creating and replacing old cells with new cells over the years. Both scenarios quite literally make up the Ship of Theseus experiment, and there is no real wrong or right answer.
What makes you "you" is continuity of conscious.
Additionally any form of portal that is not an Einstein Rosen bridge, is likely working by deconstructing your body at the origin and reconstructing it at the destination ala Stargate.
I'm not sure why people are using the ship of theseus analogy. It doesn't really fit with what op is saying.
Instead of replacing parts of a ship like theseus or our body replacing cells, this would be like the creation of two different ships with the same blueprint and materials and then bombing one and then claiming they were always the same ship.
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