Electromagnetic charge is an indestructible element of elementary particles, it cannot be exchanged.

Yes, it can be exchanged. Charged bodies exchange with charges ;-)

More seriously, it is all context-dependent. For example, take a Hydrogen atom and scatter a charged particle from it. If the projectile velocity is relatively small ($v\ll v_0$), then an adiabatic approximation (Born-Oppenheimer) is good to study the target. But if $v\gg v_0$, then the first Born approximation works fine and you can, in principle, see $|\psi_{nlm}({\bf{r}})|^2$. For $n\ge 2$, there are "free charged sub-clouds" hanging in space with the charges less than unity, like this, for example:

The elastic cross section is expressed via those clouds: the scattering amplitude is a sum over each sub-cloud. (The scattering is elastic with respect to atomic transitions $|n,l,m\rangle$, but it is in fact deep-inelastic and inclusive with respect to soft photon emission during scattering.)

If you take a free electron at rest and transfer some momentum ${\bf{q}}$ to it, it not only starts moving, but also emits some soft electromagnetic waves, which can be received far away. In other words, a charge as an interaction property of the electron, can be felt **far away**, so the notion of charge is not limited to Faraday's law of electrolysis (conservation of charge). Similarly for quarks. It is context dependent and resembles the atomic picture.