If you straight up replace your brain with another copy, made of different materials (silicon instead of carbon, machine instead of organic), even if it had the exact same data, it obviously wouldn't be you.
To who would it obviously not be you? If it were such an exact copy then people would only know if told, though, perhaps the scars would give it away?
If someone made an exact copy of you down to your memories, would your point of view shift to that of that copy? Will you start living his new experiences even as the both of you become separated? From his point of view, he either just woke up or he thinks he's been you the whole time and he just moved slightly away from where he was, but he would believe he was you. You, of course, would also believe that you are you. Still, you got two different people from that point forward, and from a continuity standpoint, only one of you has been existing and one has false memories (if only from a record keeping viewpoint).
However, if you ceased to exist at that same moment, could you say that it doesn't matter to you, because you still live in the form of a copy? I'm sure the copy is perfectly happy thinking that, but for you, the recording VCR that is life just suddenly hit STOP.
So there is merit to the OP. What if the change is gradual? First, parts of your brain could be replaced (maybe some of the empty memory space that hasn't been filled yet, although likely there is no such thing). Even if that caused you amnesia, would you still be you? Are people who suffer brain damage and are so changed still themselves? What if the amnesia parts were quickly replaced by something artificial so that, outwardly, no change occurred in your personality?
Once that synthetic part of your brain is implanted, and assuming it is as capable of the organic one to learn and experience more (could have increased Ability Scores or something, longer life without failure, etc), at some point you replace the remaining organic part with a synthetic part... what then? Is there something inherent to the organic part that "sparks" life? Since we're assuming we managed to duplicate it for discussion purposes... well, the answer in this hypothetical scenario would be no.
But anyway, this whole "losing half of your brain" is a bit extreme, and it could be argued that people that have that happen to them don't remain the same person. What if it was done in 5% increments, small enough not to notice? A bit of memory gone here, nicely backed up tho, then some motion control replaced, until it's fully synthetic, and as the OP asks, you're awake the whole time? Maybe super magic Star Trek nanites could do this without compromising the whole system.
I guess the point of this is, is this transition any different than just removing you and switching you with a duplicate?