This line of inquiry is generally referenced as the Ship of Theseus argument. The underlying philosophical question about identity holds even for inanimate objects whose parts are replaced.

But, this argument has also been extensively reapplied to brains, bodies, and minds as well. The Chinese Room thought experiment is one common reference for this, where a functional system is trained to translate Chinese texts but framed in a way to cast doubt on whether there is any understanding of Chinese.