Yes, the fundamental interactions that drive magnetism are completely and totally quantum mechanical and have absolutely no analogy in classical physics. They derive from some subtle interplay of Fermion statistics, Pauli exclusion principle, quantum tunneling and Coulomb repulsion.
Unfortunately this makes it extremely difficult to make any explanation of magnetism from a classical perspective. Actually there's some interview Feynman did where he basically goes on a bit of a tirade over this exact question because the interviewer wants a simple answer and Feynman basically says he can't give it to him. I'm sure it's on youtube somewhere.