no star systems with planets the size of Earth hosting moons the size of Earth have been discovered. However, it's worth noting that the search for exoplanets is an active and ongoing area of research, and new discoveries are made regularly. Therefore, it's possible that such systems could be discovered in the future.
Regarding the habitability of such moons, several factors would come into play. The habitability of a celestial body depends on various conditions, including its distance from the host star, its composition, atmosphere, and the presence of liquid water, among other factors. Additionally, the presence of a large moon could influence the habitability of the planet it orbits.
If a moon the size of Earth were to exist in a habitable zone around a suitable star, it could potentially possess an atmosphere, liquid water, and the necessary conditions to support life. However, it's important to note that many other factors would also contribute to habitability, such as the moon's geological activity, magnetic field, and the presence of organic compounds.
Given the complexity of these factors and the current limits of our knowledge, it is challenging to make definitive statements about the habitability of such hypothetical systems. Further advancements in astronomy and exoplanet research may provide more insights into the possibilities of such systems in the future.