Ts... Is so sad...
Would it really be the same person? If you put the memories of the original into another body, even though it is only a copy.
Would you still act the same around it?
This is a question that has been asked for thousands of years (no really, it was mostly about reincarnation and putting someone's soul in another body though), and there is no real "true answer". The ship of Theseus was partially based on this issue which infamously has no answer either.
Basically it comes down to whether you, or whomever is the observer, feels is right. The flipside is "if it walks like a duck, acts like a duck, and quacks like a duck; it's a duck" being basically "close enough, why care about the details?" And you kinda just have to decide whether it matters or not ...