This topic can be controversial at times, and as with many such things, there are multiple perspectives. Some of the discussion on this goes back to Edward Gibbon, author of the famous Decline and Fall of the Roman Empire, but this is an observation people have been making since antiquity.

Many scholars who have looked at this issue have noted the parallels in terms of practice between pagan religions and the Christian cult of the saints. Veneration of images and sacred objects, devotional practices, setting aside of certain days of the year, processions, and expectations of mutual obligations between the saint and the believer are all parallels between the two practices. As far as the nature of the cult practices, and how their practitioners perceived them as functioning, there are some clear similarities. We have some evidence, for example, of Christian saint devotees trying to "punish" saints for not fulfilling prayer requests by ritually humiliating their images. This practice was widespread enough that it was condemned by the Council of Lyon in 1274. Pagan, Jewish, and Muslim observers have pointed to such commonalities since antiquity up through the middle ages and beyond, and St. Augustine of Hippo may have been aware of such criticisms. In his Civitate Dei, Augustine does devote some space to distinguishing the miracles of Christian saints from those of "demons," a term he uses to refer to divinities other than God or the heavenly host.

Sociologically, both pagan and Christian cults have played similar roles in the lives of believers by creating a direct conduit between the heavens and the earth. One of the signature works on this topic, Peter Brown's The Cult of the Saints: Its Rise and Function in Latin Christianity, does note some differences. Brown finds evidence that pagans in late antiquity found the Christian reverence for saints strange in certain respects. Christian saints, in contrast to pagan gods and heroes, were not heroic, divine, or semi-divine figures like Zeus or Hercules, but people. They were exceptionally holy people who enjoyed a unique access to divine intercession, but they were not by their nature different from human beings. This helped create a critical linkage between the faithful and the heavens. Brown specifically challenges the idea that the cult of the saints was some kind of vestigial holdover from popular religion that the church merely accepted as the cost of doing business; it was observed and cultivated both by believers and by the leadership of the church itself. It wasn't just some sort of superstition that held on, it was widely revered and accepted at different levels of society and religious practice. The Christian cult of the saints, in this sense, developed in a Christian context to serve Christian ends.

This being said, there is some evidence of Christian appropriation of pagan spirituality in certain respects. The Pantheon in Rome, for example, was reconsecrated as a Christian church, a fairly powerful case of adopting a sacred site for new purposes. In terms of saint cults, there is some evidence from different points in the church's history of adopting pagan deities for Christian purposes. In Anglo-Saxon England, Christian chroniclers continued to trace the lineage of rulers through the Germanic god Woden, now re-imagined as an exceptionally skilled and important human. This doesn't get directly to the issue of saints, but it does show that Christians have been willing to adopt pagan tropes and types when they are useful. In more recent history, there is some evidence that the famous Marian apparition of Our Lady of Guadalupe in Mexico may have been a Christian adaptation of an earlier pagan cult practice: prior to the Spanish conquest, there was already a site sacred to the goddess Tonantzin on the hill of Tepeyac, and there are some similarities between the attributes of the Virgin of Guadalupe and other depictions of the Virgin in Spain.

All of this is to say that while there is no memo from the Vatican that went out saying "let the peasants keep the old gods so long as they call them saints;" the development of the cult of the saints has involved both the church hierarchy and the actions of believers. It has elements both of organic development and some top-down management.

Readings

Peter Brown, The Cult of the Saints: Its Rise and Function in Latin Christianity

Robin Lane Fox, Pagans and Christians

Kevin Madigan, Medieval Christianity: A New History

Robert Bartlett, Why Can the Dead do Such Great Things?: Saints and Worshippers from the Martyrs to the Reformation

Craig R. Davis, "Cultural Assimilation in the Anglo-Saxon Royal Genealogies"

Ascensión Hernández de León-Portilla, "Guadalupe de Extremadura: el culto y la historia"

Rodrigo Martínez Baracs, "Orígenes del culto en el Tepeyac"

William A. Christian, Jr. "The Spanish Shrine"

You have already received an excellent answer from u/CrankyFederalist, but I really recommend reading this older answer by u/thejukeboxhero.