In my opinion that's part of what makes it funny, and is also a core issue in the Ship of Theseus problem -- who's to say how much it can be changed and is still "the same thing"? Maybe the thing that was replaced was very small but was very important to the identity.

That’s not what the Ship of Theseus is. The thought experiment is: if you replace 100% of the original object then is it the same object. Not like, if you change the batteries in the remote is it the same remote.