How does Jupiter have two magnetic South poles and one magnetic North pole? Magnetic tripole?

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?

The magnetic fields of stars and planets are modeled by the dynamo theory, not by bars of magnet.

In physics, the dynamo theory proposes a mechanism by which a celestial body such as Earth or a star generates a magnetic field. The dynamo theory describes the process through which a rotating, convecting, and electrically conducting fluid can maintain a magnetic field over astronomical time scales. A dynamo is thought to be the source of the Earth's magnetic field and the magnetic fields of Mercury and the Jovian planets.

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Dynamo theory describes the process through which a rotating, convecting, and electrically conducting fluid acts to maintain a magnetic field. This theory is used to explain the presence of anomalously long-lived magnetic fields in astrophysical bodies. The conductive fluid in the geodynamo is liquid iron in the outer core, and in the solar dynamo is ionized gas at the tachocline. Dynamo theory of astrophysical bodies uses magnetohydrodynamic equations to investigate how the fluid can continuously regenerate the magnetic field.

You ask:

What causes Jupiter to have two different magnetic South poles and one magnetic North pole? 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)

Particularly for Jupiter there is already a dynamo model proposed:

Jupiter's magnetic field is generated by the convection of liquid metallic hydrogen in its interior. The transition from molecular hydrogen to metallic hydrogen as temperature and pressure increase is believed to be a smooth one. As a result, the electrical conductivity in Jupiter varies continuously from being negligible at the surface to a large value in the deeper region. Thus, unlike the Earth where the upper boundary of the dynamo—the dynamo radius—is definitively located at the core-mantle boundary, it is not clear at what depth dynamo action becomes significant in Jupiter. In this paper, using a numerical model of the Jovian dynamo, we examine the magnetic energy spectrum at different depth and identify a dynamo radius below which (and away from the deep inner core) the shape of the magnetic energy spectrum becomes invariant. We find that this shift in the behaviour of the magnetic energy spectrum signifies a change in the dynamics of the system as electric current becomes important.

So it is not a simple bar of magnet model that could explain the magnetic field of Jupiter, it depends on the fluid dynamics of it core. Since magnetohydrodynamics is based on electromagnetic theory, I do not think there can be a problem to model two surface south poles.

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).