first, as one responder noted, not every cell replicates at exactly the same time...
Analogy... if you understand how computer storage devices work... there is a method which uses redundancy to ensure that information survives the loss (and replacement) of a component device. Example, consider 5 disk drives working in parallel (like brain cells)... we store 4 bits of information (bits, bytes, blocks, whatever) onto 4 drives and on the 5th we store the sum of the XOR function of the other 4 drives... ( 0 XOR 0 yields a 0, 1 XOR 1 yields a 0, 0 XOR 1 yields 1 ). Now, should one of the 4 drives fails, we swap it for a new drive (cell replicates) and rebuild the missing information by simply doing the XOR on the remaining 4 drives and writing the sum on the new drive. In the cell world that would be making the neurons connect with the cells not replicated.
How do we know the analogy holds?.. remove a piece of your brain... if it is big enough on in the right area... your personality might/will change... certain memories will be forgotten. That would be like removing more than 1 disk drive from the 5 drives. It survives the loss of 1 drive, but not 2 or more.
Second, how do you know it DOESN'T change your sense of self as cells replicate (as you age)? Time definitely changes your memories as it makes new memories... but it also might change your memory all on its own.
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