Elementary charge of electron and elementary magnetic moment of electron? There is an elementary electric charge that is equal to the electron charge. Beside this electric charge, the electron has also a constant value of its magnetic moment. Is there an elementary magnetic magnetic moment, and if yes, is it as fundamental as the electron magnetic moment?
 A: Electric charge is both conserved and quantized, so it make sense to define an elementary electric charge $e$. Every free electric charge ever observed is some integer multiple of this value.
(Quark electric charges are quantized in units of $e/3$, but quarks are never observed as free particles, so in practical terms it makes sense to define the elementary electric charge as $e$, which is easily observable as the electric charge of the electron, proton, and many other particles, ions, ….)
Magnetic dipole moments of systems are neither conserved nor quantized in the same way, so there is no way to define an elementary magnetic moment in the same sense. For example, the intrinsic magnetic moments of muons and electron are fixed but very different, but this does not prevent $\mu\rightarrow e\nu\bar{\nu}$ decay.  Similarly the initial and final state can have different magnetic moment for any atomic transition where the electrons flip spin or change orbital angular momentum.
The magnetic moment of a particle (or any system) depends on factors such as electric charge, spin, angular momentum, mass, and structure.
One can use the electron magnetic moment, $\mu_e=−9.2847647043(28)\times10^{−24}\,\textrm{J/T}$, as unit for magnetic moment, e.g.
$$\begin{align}\mu_{proton} &= -1.519\times10^{-3}\mu_e \\ 
\mu_{muon} &= +4.836\times10^{-3}\mu_e \\
\mu_{deuteron} &= -0.466\times10^{-3}\mu_e \\
\mu_{Hydrogen\,atom} & \sim 1\mu_e\\
\mu_{Holmium\,atom} & \sim 11\mu_e
\end{align}$$
but this is in no way more fundamental than any other choice of unit for magnetic moments, e.g. J/T, Bohr magneton, nuclear magneton, …
