There are three ways magic tends to function in religion and mythology that Fantasy fiction very rarely emulates:
Sympathetic magic, that is, the idea that conceptually similar objects have a magical link and that manipulating one in a ritual way can affect the other. The archetypal example is prolly effigies and voodoo dolls: by harming a representation of a person you wishing harm on that person, but it shows up a lot in folk medicine as well.
Contagious magic, that is, the idea that objects that come in contact with one another influence each other in magical ways that linger even after they are separated. Using a lock of someones hair to cast a curse upon them or an object owned by someone in a seance is example. A non-mystical example of that would be the importance people place on objects owned by celebrities.
and third… imma call this divine magic, even though that’s only part of what im talking about. Basically it’s when you ask a god or supernatural being for help. The genie in the lamp would be an archetypal example, but it’s worth mentioning that in the original fairy tales, that’s only one of many ways Djinns could be captured and made to perform a specific task for you.
The common thread in folk magic is that it tends to be based on… well, on magical thinking. On the sort of things that intuitively FEEL logical before you think rationally about it: “Need a cure for impotency? Try horn powder, horns are stiff and rigid, surely they must contain some ‘horny’ quality that you could extract somehow.,” “I don’t wanna wear that shirt, its been worn by a murderer! Yeah, its just a store-brought shirt, but what if there’s evil all over it?”, “In order to do something you need a doer, so if I want to do something I can’t, I need to call forth a being who can.”
If you are gonna design your own magic system, I think the key is to create similar alternative paths of “logic.” Cartoon physics could be seen as a form of magic: by knowing how the world operates, a skilled practitioner can use it to their advantage. I cast “summon anvil”
An example from my own work is what I jokingly call “the postal system” (Oneiromancy in the actual text), which is based on opening letters. Its content and how far the letter has traveled determine its effect (Chainmail protects the user, “Dear Johns” hurt a target and so on). Feel free to steal that concept if you wish, I’m not too protective of it.