So, by your reasoning, it would not be the same person. Transferring memories around like this does NOT transfer people around like this, since the particles that make up their bodies and brains are entirely different. The fact that they "feel" that they are the same person in a new body is irrelevant.
They also sometimes spin people up inside virtual reality environments, with no body at all. I guess in that case the person isn't actually there at all. There is no person, there is just a computer program running as if it were a person. And when that hard drive gets inserted into a new body, including the memories of its activities in virtual reality, all that has really happened is that "someone else" who has been in a coma until now just woke up with a convincing set of false memories.
This.....doesn't seem to suit our intuitions. The general reception to this scifi is that the "person" is in that hard drive, and the aptly named "sleeve" (body) is not part of the personal identity. If we accept your analysis of star trek then we must reject this analysis of Altered Carbon, and vice versa. These two intuitions for what constitutes a "person" are directly contradictory, even though they both seem to make sense in context.
You are not a ghost riding a corpse, you are your body.
Your brain is part of your body, and your mind is an emergent property of activity in your body... Your brain being a part of your body after all ;)
The People* in Altered Carbon aren't Humans.
They are memetic parasites.
They are still people, and by law personhood follows the movement of the DHF Digital Human Freight.
When a mind in AC is backed up, and loaded into a new body the thinking isn't done by the stack in their neck, that is a storage cache and transmission device. An interface between brain and machine.
The actual work of thinking is conducted by the brain of the sleeve, which is why drugs still make them high.
Also the body retains it's own reflex arcs and muscle memories.
In the books there is a character training up a young sleeve to be a great surfer for a wealthy person, he's training up the muscle memories and so on, and when the sleeve is adult and knows how to surf at a champion level, he has to return it.
At the end of the first book when Takashi has to give Rikker's body back and he's in a new body, he doesn't feel the same way about Ortega because the hormone mix is different. Pheromones aren't ticking the same neural switches.
Re-sleeving replaces the higher functions of the brain, but not things going on in the lizard brain or the "gut".
There isn't some magical soul energy in AC.
The stack copies the current brain, then wipes it through some techno-electro-stimulation wizardry.
And then injects the new data.
When they run people in virtual they don't just spin the stack up, they create a virtual body/brain, and spin the mind up on that like a VM. Which is why in the book they spin Tak up in a vitural female body complete with virtual PMS in an attempt to keep him in an unfamiliar situation.
*This is why the people are protesting, it's not because they are annoyed about "the dead testifying" rather they are protesting because every time they get arrested for anything that requires being held for any period of time, they are "spun down", and their bodies are rented out by the state to people who want a fresh body for the weekend/month/length of time they are in custody.
For the religious people who see the mind wipe aspect as death, they see all crimes being punished by death.
There are more minds then there are bodies,
if you are old and wealthy, but not so wealthy that you can afford a custom sleeve like the meths get, then you hire a nice fresh sleeve that some poor had to leave when they went to prison for a hit and run.
Oh? they're on stack for 50 years? well that's plenty of time for you to use their early 20s body that they kept in great shape. Then when they get out, you've been using their body for 50 years, and have put a lot of miles on it, and they're suddenly 50 with no money for a nice fresh body. If they even get their original body back, maybe it died, and they get whatever is left in stock. Like the "old lady" in the first episode... and she wasn't even a criminal.
Mind Transfer in Altered Carbon is not soul transfer.
It is a terrifying dystopia.
You are a gestalt of body and mind.
Not a collection of memories.