In a single year, 98% of the atoms that comprise your body are replaced. IRL our consciousness is clearly not tied to the "stuff" we are made out of - even if it is rooted in the physical world. You could sub out all of the atoms you are made of for different atoms, and (since all atoms are identical of a given element/ion) it would be you with no perceptual gap. Hell, given humans can lose limbs (or even parts of their brains) and seem to report being the same person, a perfect swap likely isn't even needed - just one which is "good enough."
Turning to the Trekverse, the only thing that really matters is if the story treats Picard as the same person. All indications are he is. Everyone surrounding him takes for granted he's Jean Luc, and he has a first-person experience right up until the moment of his (old body's) death.