But is it really the same person? We assume that because we see the same face, hear the same voice, but is that too superficial an analysis? For that matter, am I the same person I was 20 years ago? Or have I become a different person who retains many memories and habits of the person I was then, but has also lost a lot of who that person was and gained other attributes? Identity is not a simple thing to define. It's unwise to leave your assumptions unexamined when dealing with a situation outside your past experience.
And the situation here doesn't fit your analogy very well. This is a different positronic brain, constructed to house a different personality, with its experiential memories wiped and replaced with a copy of a copy of the memories from the original positronic brain. Also, brain damage tends to take elements away; this new brain has added abilities and enhanced performance. It's kind of the reverse of brain damage.
The problem is that we don't have a hard and fast definition of 'identity'. Soong's character hit it best when he mused that, given the time and resources, he could use nanites/nanoprobes to pull a Ship of Theseus on himself for maximum assurance regarding identity. Even then, though, there's not a way to know whether continuity of consciousness would survive.
It's the continuity of consciousness that's the real problem here. The truth is, we simply don't know how any of these issues can resolve themselves in real life. Maybe consciousness is malleable enough to survive the sort of indignities one would experience in Trill joining, uploading, resurrection, or even the dematerialization and rematerialization of transport.
But then again, it may not.
And how would we ever know?
Unless the Betazoid or the Q have some special ability to discern that we could take advantage of to determine it, the answer can't be found.
There's a book called "The Philosophy of Star Trek" that tackles this issue vis a vis the Trill and transporters specifically, along with other crazy identity weirdness in Trek, and I like a concept they brought in as a resolution called something like 'closest continuing individual' or something to that effect (unfortunately I can't find my copy at the moment to verify, can someone help me out here?).
If we can accept that the person that comes out the other end of a transporter is the same person who went in, or that James Kirk is still Kirk even after he's been split into two people with different personalities and reintegrated, or that Spock would be the same after dying, being reconstituted and having his katra restored to his body, then I see no problem expanding the concept to include what has happened to Data as a closest continuer to the Data of old.
Is he an unbroken consciousness? No. But then, what happens when he goes through transport? When he's shut down? When you flip his switch? When Q turned him (temporarily) human? We accepted those changes. Why not this?