The confusion is about the Weyl limit--- a massless Weyl fermion can be charged. All the fermions in the standard model are charged Weyl fermions.
The Weyl fermion that can't be charged is the massive Weyl fermion. The reason is that the mass term in the Weyl reduction mixes up the field and its conjugate, so it isn't phase-invariant under multiplying the Weyl field by a complex phase. This type of mass, which is incompatible with charge, is more often called a Majorana mass in the literature, because it is easier to derive as the real part of the Dirac equation in a real basis.
The fact that Weyl fermions can't have mass is important--- it is the reason we see Weyl fermions in nature--- if they could be massive, they would be Planck mass massive. Instead, they are only Higgs-scale massive.
A pair of massive Weyl fermions with the same mass can together be charged, with the charge symmetry rotating one into the other. So a pair of Weyl fermions can also be massive independent of the mass, and the reason is that this is what the Dirac equation is.
The answer to the title question, about the equality of charges, at least from the theoretical standpoint is found here: What is "charge discreteness"? . Anna v. has given the experimental reason.