Magnetism isn't made of anything. It is a field, and it's one your can interact with directly.
Compare that with, say, table salt. "What's table salt made of?" Uh. It's made of atoms entirely unrelated to salt, like sodium and chlorine. "What are atoms made of?" Uh. They're made of tiny electric particles zipping around at relativistic speeds but bound together by forces like magnetism.
Richard Feynman does a better job than me of describing this exact problem in one of my favorite physics videos of all time:
Why do you say that? Do you think there's anything fundamental?
We actually were convinced atoms were fundamental for a while, until we stumbled upon evidence they're not. Now we think fields are fundamental, and as far as I know, we have no reason to believe otherwise.
Contrast this with atoms: there are things atoms alone cannot explain, like the Zeeman and Stark effects. There must be something more fundamental going on. (Spoiler: It's fields.)