The world's person

Confusing the being with the being currently associated with the title. It is a form of equivocation .

The world's oldest person keeps dying over and over again. 

This is incorrect. If we say that "the world's oldest person" is referring to a specific person, then it is incorrect that the specific person has died more than once. If we say that "the world's oldest person" is referring to the person whomever happens to hold that title at the time, then it is incorrect because each person who held that title died just once.

As an attempt at logic, the sequence is equivocating between "the currently oldest person" and "all those who were the oldest when they died".   The "this person" could be taken as either, depending on how one wanted to argue.

As an attempt at humour, the sequence is essentially an expanded pun ... not a word with double meaning; rather the concept of "world's oldest person" having double meaning.