Generation of magnetic field of Earth is relatively well-understood
It is? I do not think that is true. We have a theory and an idea, but there are rather large holes and unknowns in both. Dynamo theory is, yes, the basic idea but it's fraught with issues so I would not say we have a solid grasp of this quite yet.
What causes Jupiter to have two different magnetic South poles and one magnetic North pole?
Have you ever looked at the magnetic field topology of the Sun during solar maximum? It looks like lots of little dipoles, quadrapoles, octapoles, etc. on the surface with little semblance of something like a global dipole geometry. However, the convenient thing is that the higher order multipole moments fall off faster with radial distance than the dipole field, so at large distances the magnetic field starts to come closer to something like a dipole (it's still extremely messy).
The Jovian magnetic field is similar to the sun in that it exhibits significant higher order multipole moments like the quadrapole and octapole moments. This can result in rather unusual looking field geometries compared to a simple dipole.
Does this mean two magnetic South poles have the same magnetic North pole? (I'm assuming so because otherwise I see a weird monopole appearing even though they are not forbidden by nature)
No, there's no monopole moment of which we are aware. When higher order moments become significant, the field geometry can become very complicated, however, giving us what Juno has found.
If one can approximate Earth's magnetic field with an enormous bar magnet, how should one approximate Jupiter's magnetic field to have a simpler picture?
Technically, you can only approximate Earth's magnetic field as being like a dipole bar magnet at large distances from the surface. There are large deviations from a strict dipole geometry like the South Atlantic Anomaly, which result from multipole moments like on Jupiter and the Sun. Again, these decrease in magnitude much faster than the dipole moment, so far from the source the global magnetic field of Earth kind of looks like a dipole. It is technically called a stretched dipole because the supersonic solar wind distorts it, but from ~1-6 Earth radii from the surface the field geometry usually looks like a dipole (except during a geomagnetic storm).