Transition of electric charge to "magnetic charge" when $\alpha$ gets >> 1 in QED?

Question

I`ve just learned that electrically charged particles and magnetically charged monopoles in QED are S-dual to each other such that it depends on the value of the fine structure constant which of the two is the ligher (and more fundamental) one and which is heavier (and more complicated).

Whereas I think I understand reasonably well how and why the roles of the light and the heavy particle concerning their mass get interchanged as the coupling sonstant increases from $\alpha << 1$ to $\alpha >> 1$, I fail to grasp how the electric charge is turned into a magnetic charge in the course of this (hypothetical) process.

Should an electrically charged particle, such as an electron for example, not be fundamental different to some kind of a magnetically charged (hypothetical) monopole (I mean two different fundamental particles)? Non of the different "pictures" I`ve seen so far to "visualize" S-duality explain what happens to the kind of charge ...