Charge is a quantity which arises from Noether's theorem, due to continuous global symmetries (up to a total derivative) of a Lagrangian, and as such we have many types of charge, other than electric. For example, consider the Dirac Lagrangian,
$$\mathcal{L} = \bar{\psi}(i\gamma^{\mu}\partial_{\mu}-m)\psi$$
which describes fermions. It is invariant by a change of phase, i.e. $\psi \to e^{-i\alpha}\psi$, which has a conserved current, namely,
$$j^{\mu} = \bar{\psi}\gamma^\mu \psi$$
The conserved quantity, i.e. Noether charge, arising from the symmetry is the integral over all space of the zeroth component, therefore
$$Q = \int \mathrm{d}^3 x \, \, \bar{\psi} \gamma^0 \psi = \int \mathrm{d}^3 x \, \, \psi^{\dagger}\psi$$
The quantity $Q$ is indeed electric charge. In addition, when the field $\psi$ is quantized, and expanded as a plane wave with operators as Fourier coefficients, it can be shown that $Q$ also has the interpretation of particle number for fermions. In classical electromagnetism, charge also determines the magnitude of the effect of magnetic and electric fields on charged matter, via the Lorentz force relation,
$$\vec{F} = q\left(\vec{E}+ \vec{v} \times \vec{B}\right)$$
The elementary charge $e$ also plays the role of the coupling constant in quantum electrodynamics, which roughly determines the strength of an interaction term, namely,
$$\mathcal{L}_{\mathrm{int}} \sim e \bar{\psi}\gamma^\mu A_\mu \psi$$
corresponding to an interaction vertex involving a photon, positron and electron. In addition, the coupling $e$ actually depends on a scale, dictated by the beta function (to one loop order),
$$\beta (e) = \frac{e^3}{12\pi^2}$$
So the idea that there is a single $e$ is wrong; from the beta function, assuming no additional corrections alter its nature, we may deduce the coupling increases with increasing energy scale.