What is our solar system called?
The name of our planet is the Earth. The name of our moon is the Moon. The name of our solar system is the Solar System.
I capitalize them, because when used as names, they are proper nouns. This also helps us distinguish between the planet Earth and earth (meaning soil), between the Earth's Moon and moon (meaning the natural satellite of a planet), and between our Solar System and any other solar systems (since any system containing a star and a planet or a planet-forming disk can be called a solar system.)
The Milky Way is the galaxy that contains our Solar System.