I should point out that, under the constraints of the 20-year-rule of this sub, if John Paul II indeed starts this trend, he would be, for our purposes, a one-off phenomenon. He would remain pope until 2005. The short answer is that no, the papal celebrity factor doesn't really exist until the 20th century.

Before the 14th century, priests were asked that they prevent parents from giving their children ridiculous (and especially pagan) names, but this was not really universally observed. With the rise of the morality play and hagiographies for mass consumption in the 14th century, names began to take a turn—not towards the reigning pope but towards great saints. Spain and Greece even began to use names like Christ and Jesus for young boys. Catholic Europe doubled down on this with the Reformation in the 16th century, perhaps as a show of Catholic identity. (Protestants would replace saint names with the names of Old Testament figures; hence the stereotype in American media of good Protestant farmboys being named Obadiah.) In the Catholic world, great saints, especially local saints, would reign supreme.

In the Italian city of Padua, for example, Anthony has historically been the most popular name for boys, and Maria for girls. Maria, as a variant of Mary, is uniformly popular in Catholic circles---especially in Mexico, where she doubles as the subject of the Guadalupe apparitions---while Anthony was the name of Padua’s famous patron saint, Anthony of Padua.

A quick perusal of baptismal records in one parish of the United States gives an example of this pattern: Nearly every person in the Holy Trinity Church baptismal register in 1793 were given names which are easily associated with saints; though it is possible that they were named after family, those family bearing names like Catherine hearken back to the days when Catherine of Alexandria was one of the most popular saints of Medieval Europe. None are named Pius, the regnal name of the pope at the time. (His birth name was Giovanni, or John, a popular name no matter where you are, so I guess that could be inconclusive.)

I can’t find a source on this, so here’s my best guess at what made it change: It would take the rise of mass media beginning in the late 19th century, the dispute (and victory) of the pope-focused ultramontanists around the same time at the First Vatican Council, and the disruption of the Second Vatican Council in the 1960s for a figure like John Paul II to use celebrity status to attempt to unify a church undergoing several crises.

Baptismal names appear to follow from what the churchgoers care about; since at least John Paul II, they care a lot more about the reigning pope.

Minello, Alessandra, Gianpiero Dalla-Zuanna, and Guido Alfani. “The Growing Number of given Names as a Clue to the Beginning of the Demographic Transition in Europe.” Demographic Research 45 (2021): 187–220. https://www.jstor.org/stable/48640777.

“Names, Christian.” New Catholic Encyclopedia. 2003. Detroit: Thomson/Gale.

Reuss, F. X., and Thomas C. Middleton. “Baptismal Registers of Holy Trinity Church, Philadelphia, for 1793-4-5.” Records of the American Catholic Historical Society of Philadelphia 22, no. 1 (1911): 1–20. http://www.jstor.org/stable/44208141.