What prevents us from considering the magnetic dipole of subatomic particles to be as fundamental as the charge After the discovery that subatomic particles have a magnetic field, explanations were sought. Ultimately, it boiled down to the fact that electric circuit currents in the particles are responsible for this.
The question is as simple as it is formulated in the title.
 A: $\leadsto$ The fact that magnetic moments of particles (the strength with which their spin couples to external magnetic fields in an effective Lagrangian) are not multiples of each other.
This is in sharp contrast to "electric charge quantization": the fact that isolatable particles have an electric charge which is a multiple of an "elementary charge", e.g. the electron's. There is a host of theories, some involving magnetic monopoles, attempting to explain this, and GUTs (which are unconfirmed) explain the fractional charges of (nonisolatable) quarks, still remarkably simple rational fractions.
Simple elementary particles (e,μ,...) have a simple value, $-2\mu_B \vec{s}/\hbar$, dictated by the Dirac equation they obey (so the Bohr magneton scales down with the mass of the particle, so the muon magneton is $\mu_B m_e/m_\mu$); but quantum corrections modify this value to an anomalous value you'd find in the PDG. (Electric charges do not get modified this way, basically by gauge invariance,  not protecting magnetic moments for  recondite reasons.)
Composite subatomic particles, like baryons (p,n,...) have freaky, not obviously rational values w.r.t. each other's, prominently displayed in the PDG. They feature in the effective Lagrangian for them, coming from impossible-to-really-calculate (in our lifetime) QCD corrections. It would be meaningless to call them "fundamental" in any sense, even though elementarity is a conceptually fraught idea...

Response to comment questions
Not sure how to expand in general. The wikipedia article linked covers a lot of ground. I only understand it through the math. Speaking about engineering alignment of charges and currents in atoms is deprecated (they are decidedly quantum mechanical systems, where classical imagery fails systematically and jubilantly!).


*Yes, all discussion is about intrinsic magnetic moments, never, ever induced. However, if you went to the PDG, you'd  see it also has a magnetic polarizability, as well, about which I know little.


*I know little about alignments of multi electron or multi magnetic systems, so I didn't understand the question. The huge (!!) intrinsic magnetic moment of the neutron is due to its peculiar (internal) alignment of quarks, its spin-flavor(-color) wavefunction, and the quark model estimates it creditably. The unit is the nuclear magneton (for charge 1, but the neutron is chargeless!!)
