In general, only the Pope is allowed to wear a white cassock. However, in very hot countries all members of the clergy are allowed to wear so called tropical cassocks in order to protect themselves from the sunlight. Hence, Cardinal Voiello is wearing a white cassock (instead of his usual black one) trimmed in his traditional cardinalatial scarlet.
The swimming pool in the Papal summer residence in Castel Gandolfo was build in the early 1980s for pope John Paul II who was very athletic (much like Pius XIII in the series) and did lots of sports like swimming, skiing or playing soccer.
While being told of the Blessed Yuans, the scene cuts to her village which is the same set used in West World where Doris meets the little girl sitting on the well wall.
The tradition of press conferences with the pope on the Papal plane was introduced by the polish Pope St. John Paul II.
Pius' XIII dislike of Castel Gandolfo is an allusion to Pope Francis who also has no summer vacations and doesn't visit Castel Gandolfo at all.