"Personality drift" while you sleep probably does happen, but in small degrees. You don't think exactly the same as you did 10 years ago. People have been knocked unconscious and woken up with different personalities, so it's not like people always wake up with conscious continuity.

Sleep and unconsciousness are more accessible means of exploring these thought experiments than fantasy teleportation devices, but many of the considerations apply. If you're a strong materialist, then the notion that "consciousness" is special is silly: any body that has your thoughts is "you" and multiple "yous" is fine, they should diverge as each copy has unique experiences.

On the other hand, many people are not materialists at all. Many believe either explicitly in a supernatural soul, or in a more ineffable "higher consciousness" that science has yet to reliably demonstrate. For these people, continuity of consciousness has severe implications.

If a person has a brain injury and wakes up as a totally different person, what happens to their soul? (I'm a materialist, so I dunno. Just pointing out that the question does have meaning to people.)