Do the electric and magnetic components of an electromagnetic wave really generate each other?
No they don't. Like Andrea said, they're two "aspects" of the same thing. And like you said, it's an electromagnetic wave. See the wiki article for electromagnetic radiation where you can read that "the curl operator on one side of these equations results in first-order spatial derivatives of the wave solution, while the time-derivative on the other side of the equations, which gives the other field, is first order in time". One's the spatial derivative, the other's the time derivative. If it was a water wave and you were in a canoe, the tilt of your canoe is E and the rate of change of tilt is B.
Frequently when EM waves are taught, it is said that the change in electric field causes a change in the magnetic field, which then causes a change in the electric field, and so on and so forth.
Yes, and the people who say this tend to say this is why electromagnetic waves don't need a medium. But they do need a medium. Space is the medium. It isn't a medium like water or the ground, but it isn't nothing. See LIGO and note that they're trying to detect a length-change in the arms of the interferometer. That's essentially space waving. Also see this where Robert B Laughlin talks about quantum vacuum and likens space to window glass rather than Newtonian emptiness.
and it is this charge that creates both the electric and magnetic field
IMHO you should avoid charge for now and stick to electromagnetic waves.
Is the "mutual generation" concept between the electric and magnetic components of an EM wave an actual thing?
No it isn't. Check out this Wikipedia article about Jefimenko's equations:
There is a widespread interpretation of Maxwell's equations indicating that spatially varying electric and magnetic fields can cause each other to change in time, thus giving rise to a propagating electromagnetic wave (electromagnetism). However, Jefimenko's equations show an alternative point of view. Jefimenko says, "...neither Maxwell's equations nor their solutions indicate an existence of causal links between electric and magnetic fields. Therefore, we must conclude that an electromagnetic field is a dual entity always having an electric and a magnetic component simultaneously created by their common sources: time-variable electric charges and currents".