From what I understood, the magnetic field around the Earth exists because of the rotation of the Earth. The mantle consists of molten metallic ions, which, when they rotate in sync with the rotation of the Earth, form a magnetic field outside of the planet.
The generation of an intrinsic magnetic field within a planet is actually still a big unknown and topic of great debate. The leading idea involves something called dynamo theory. The overly simplistic explanation is that ferromagnetic, molten metals move in a coherent enough manner to generate a net magnetic field that happens to be a dipole (e.g., the Sun's magnetic field is extremely complicated and not dominated by dipole terms).
Also, according to my understanding, geosynchronous orbits are those in which the satellite in the said orbit always hovers above a fixed location above the Earth.
Technically, what you are describing is called a geostationary orbit, i.e., the spacecraft stays above the same geographic location for the entire orbit. A geosynchronous orbit returns to the same position in the sky every sidereal day.
Also, I know that magnetic fields are only observed when a charge is in motion with respect to an observer.
This is not necessarily true, at least not in the literal sense of this statement. A ferromagnetic material has a net magnetic moment and magnetic field, but does not have electric currents within.
Now, for a satellite in perfect geosynchronous orbit (assuming that the ions in the mantle rotate perfectly in sync with the Earth), since the rotating ions are stationary with respect to the satellite, does the satellite experience any magnetic fields?
First, see my comments above about the difference between geosynchronous and geostationary orbits. Second, even spacecraft in geostationary orbits will see the Earth's magnetic field for the same reason a magnetometer will respond to a stationary, ferromagnetic object. For the external observer, it doesn't really matter how the fields are generated within the Earth, they exist and there is no Lorentz transformation or Galilean transformation to get rid of them.
As a thought experiment, suppose you had an infinite wire with a current flowing through it. This will generate a magnetic field that is everywhere orthogonal to the surface of the wire, i.e., the azimuthal direction. Currents are generated by the relative drift between positive and negative charges. You can transform to a reference frame where the positive ions are stationary or to one where the negative electrons are stationary, but no frame exists where both are stationary. Thus, you cannot "transform away" magnetic fields like you can electric fields.
Even for the real case, is there some observable change?
Change relative to what? The magnetosphere is not symmetric relative to the Earth-Sun line, so even for geostationary orbits, the magnetic field will change over the course of one day (i.e., geometry and strength).
There are also lots of magnetic anomalies in Earth's magnetic field (e.g., the South Atlantic Anomaly (SAA). That is, it's not a perfect dipole by any means.
