The solar system was named long before we knew that other solar systems existed. Just like the Sun is not the only Sun in the universe but we still call it the Sun.
The stars in the inner ≈10,000 light-years form a bulge and one or more bars that radiate from the bulge. The very center is marked by an intense radio source, named Sagittarius A*, which is likely to be a supermassive black hole.
Stars and gases at a wide range of distances from the Galactic Center orbit at approximately 220 kilometers per second. The constant rotation speed contradicts the laws of Keplerian dynamics and suggests that much of the mass of the Milky Way does not emit or absorb electromagnetic radiation. This mass has been termed “dark matter”.The rotational period is about 240 million years at the position of the Sun.The Milky Way as a whole is moving at a velocity of approximately 600 km per second with respect to extragalactic frames of reference. The oldest stars in the Milky Way are nearly as old as the Universe itself and thus likely formed shortly after the Dark Ages of the Big Bang.
The Milky Way has several satellite galaxies and is part of the Local Group of galaxies, which is a component of the Virgo Supercluster, which is itself a component of the Laniakea Supercluster.
The Solar System also goes by the names: The Copernican System, The Heliocentric System, and The Planetary System. There aren’t too many other names, actually, so just stick to Solar System since it’s the most widely accepted.
The name of our galaxy is the Milky Way. Our Sun and all of the stars that you see at night belong to the Milky Way. When you go outside on a dark night and look up, you will see a milky, misty-looking band stretching across the sky.