Your statement "Maxwell's equations imply that magnetic fields are due to changes in electric fields." is not complete.
A corrected statement is that Maxwell's equations imply that magnetic fields are due to changes in electric fields AND due to currents (which can be stationary):
$$
\nabla\times\mathbf{H} = \mathbf{J}+\partial\mathbf{D}/\partial t
$$
As you can see, the magnetic field $\mathbf{H}$ has two "sources": the $\partial\mathbf{D}/\partial t$ part is due to varying electric fields as you said (where $\mathbf{D}$ is the electric displacement), but the $\mathbf{J}$ is the part due to the free currents. This is why a coiled wire with a constant current running through it creates a magnetic field (without need of changing electric fields).
In the case of the electron spin, this goes beyond my areas of knowledge, but according to my limited understanding of quantum mechanics, the magnetic field comes from the stationary particle current associated with the wave-function of the electron. So it is similar to the magnetic field arising from a coil of wire with a current.
As a further related note: interestingly, Maxwell's equations apply to any inertial frame, so you could argue that an observer moving with respect to the electron will see a changing electric field (because the electron is moving), and this will create a magnetic field which apparently does not exist for a stationary observer. This is because different observers will not agree on the electric and magnetic fields separately, however they will agree on the existence of an electromagnetic tensor (which includes the electric and magnetic fields as its "parts"), and they will agree on the physical effects produced by it.