Hey! While there is more that can be said on this topic, I have written up a couple of responses to this question that address it from a couple of different angles. I've included links and summaries below, feel free to ask any follow up questions you have.

The Origins of the Cult of the Saints: Here I tackle the origin of saint cults, largely drawing off the work of Peter Brown in the Cult of the Saints: Its Rise and Function in Latin Christianity. The work is starting to age, but Brown is still the starting point IMO for anyone interested in late ancient saint cults. The main takeaway here is the discontinuities between hero and god cults and early saint cults both in conceptualization and veneration.

Cultural Syncretism and Continuities. That being said, it would be a disservice not to recognize and incorporate into our discussion the larger shared cultural vocabulary surrounding the supernatural in late antiquity within and against late ancient and early medieval Christians articulated their ideas and beliefs about saints. I also try to acknowledge the role of cultural syncretism and the diversity of experience. I provide additional context in this post.

As a bonus, here is another post that synthesizes a lot of the information above to discuss the early cult of relics, a phenomenon that illustrates both the continuities and discontinuities in the larger discourse on the supernatural in the Mediterranean world.

The TL:DR of all this is that cult of the saints developed within existing paradigms that defined mankind's relationship to the supernatural. The salient point is that nothing arises in a vacuum. While the cult of the saints was new and different, it still took root within a specific historical context and could only be articulated within --and in opposition to-- cultural paradigms that existed at the time. Essentially, early Christians drew on existing language and practice to articulate in language and practice the new thing they were doing.