The Pope is an Argentinian and he wants to keep it that way. The Pope is a Vatican citizen, but he just renewed his Argentinian national identity card and passport.
Just like any other Argentinian who is living abroad, Pope Francis will have to wait until his new documents arrive, in his case, at the Vatican's Casa Santa Marta. He specifically asked not to have any privileges that would make the process easier or faster.
Pope Francis didn't want to give up his nationality. Still, he became a Vatican citizen the moment he was elected Pope.
As the Pope gets ready to do some more traveling, with his briefcase full of values, he will of course need all his proper documents. In the past he has explained that it has a prayer booklet, a book by St. Thérèse of Lisieux, his diary and shaving razor. Soon he will also add his new Argentinian ID and passport to that list.
Ministerio del Interior Argentino