Why isn't electron spin a real rotation? When an electric current flows, a magnetic field is created.
Then, isn't the magnetic field of a magnet also caused by an electric current?
 A: 
Then, isn't the magnetic field of a magnet also caused by an electric current?

Morally, yes, although it's complicated because of quantum mechanics.
The origin of magnetism is electron spin. A crude classical model is that the electron is a tiny spinning ball of charge. In this model, the moving charge in your question is the rotating ball, and the moving charge generates a magnetic field. The more complete explanation involves quantum mechanics, where the story gets quite subtle and one finds the spin can't literally be thought of as a spinning ball. (Here is a classic meme that sums up electron spin....) But, it's not too wrong to say that the magnetic moment is due to rotating charge and often is a good place to start to get an intuition for what's going on.
Iron has a particularly large magnetic moment due to its electronic structure; it has an unpaired electron in its outer orbit, whose spin causes the iron atom to have a sizable magnetic moment (there is more to the story, in particular a complete explanation includes a discussion of the exchange interaction, but for a first pass you don't need to dig into this). In what is colloquially called a magnet (what physicists would call a "ferromagnet"), many of these atomic magnetic moments align with each other over a large area (or "domain"), and their fields add to a large combined field we can observe on macroscopic scales.
There is much, much more to say -- there other forms of magnetism (materials which respond to an applied magnetic field), the story of how the large domains get formed in ferromagnets is quite interesting, superconducting materials have mindblowing magnetic properties -- but the point of this answer is to say, you are on the right track! And there is a huge and fascinating story here to dig into, if you are interested.


Why isn't electron spin a real rotation?

A relatively straightforward argument for why electron spin is not simply the result of a literal spinning ball of charge follows from computing the velocity of a charge element on surface of the hypothetical ball. [Disclaimer: The argument I give below isn't strictly self-consistent because it is non-relativistic, as pointed out by Andrea. I think the conclusion is still ok though.]
The "classical electron radius" (which we can think of as an upper bound of the size of the electron) is $r_{e,cl}=2.8\times 10^{-15}\ {\rm m}$. The spin angular momentum of an electron (which is a spin-1/2 particle) is $S=\hbar/2=5.3 \times 10^{-35}\ {\rm m^2\ kg\ s^{-1}}$. Then the angular velocity is $\omega=S/I$, where the moment of inertia (assuming the electron is a solid sphere, though this doesn't matter for this argument) is $I=\frac{2}{5} m_e r_{e,cl}^2 $. And the velocity of a charge element of the surface is $v=\omega r_{e,cl}$. Putting it all together, we have
\begin{equation}
v = \frac{5\hbar}{4 m_e r_{e,cl}} = 5.1 \times 10^{10}\ {\rm m\ s^{-1}} = 170 c
\end{equation}
The velocity only gets larger if we account for the fact that the "true" size of the electron is likely much smaller than its classical radius (actually as far as we know the electron is exactly pointlike!) You can't have charge elements moving faster than the speed of light, so the "spinning ball" model of spin doesn't work.
A: 
Why isn't electron spin a real rotation?

The mainstream mode of physics at present is that the standard model of particle physics, a quantum field theory , describes the plethora  of data in particle physics and is accurate in predictions. All other physics models/theories are emergent from the quantum mechanical level.
The table of particles in the link depends on innumerable data for its definition, as point particles with spin.  The spin has been assigned to each particle from fits to the data, so that conservation of angular momentum holds in the standard model.
So in the existing model where elementary particles have no constituents, there is nothing that can rotate in a point particle.
