We can satisfy this curiosity with a fair amount of scientific evidence.

Of course, most regions of the brain are so densely and variably interconnected that the technical difficulty of “replacing parts” precedes the ethical consideration by many, many years. But we do have a great deal of evidence for how our subjective sense of self is affected by “losing/removing parts” of the brain. Patients are often unaware of change unless evidence for it is overwhelming, and even then are adept at healing/reconciling instinctively. It appears that this is just something brains have evolved to do.

So while the technology (and sheer artistry) required to match and “stitch” these networks is quite staggering, basically magic, it is theoretically possible that a patient could have every part replaced without recognizing any continuity errors in the chimeric stages, until one day they wake up as a completely different person.