The pope has no relation to Paul. If there is any relation, it’s a symbolic one.
This website describes the process of choosing a new pope:
Theoretically, any male Roman Catholic can be chosen. In practice, one of the cardinals is picked. Four ballots are generally conducted each day—two in the morning and two in the afternoon. Each time an unsuccessful pair of ballots has been completed, chemicals are added to the ballot papers before they are burned in a small stove. This produces black smoke which is visible above the roof of the chapel. When the new pope has been chosen, the papers are burned with the addition of other chemicals to produce white smoke. This announces to the world that the conclave has selected the new pope.
When John Paul II passed away and Pope Benedict was chosen, there were several candidates from all over the globe. John Paul II was from Poland, Benedict XVI was from Bavaria. The apostle was from somewhere in Judea. If any of them were related, it would be an almost impossible coincidence.