No, the terms "Milky Way" and "Galaxy" do not refer to the solar system. They have different meanings in astronomy:
Milky Way: The Milky Way is the name of the galaxy in which our solar system resides. It is a vast, disk-shaped collection of stars, gas, dust, and other celestial objects. When we look up at the night sky and see a band of light stretching across it, we are seeing a portion of the Milky Way galaxy. The Milky Way is estimated to contain billions of stars, including our Sun, and is just one of many galaxies in the universe.
Galaxy: A galaxy is a large, gravitationally bound system of stars, stellar remnants, gas, dust, and other celestial objects. Galaxies come in different shapes and sizes, ranging from elliptical galaxies to spiral galaxies, and irregular galaxies. Each galaxy can contain billions or even trillions of stars, along with other astronomical components. Our solar system is part of the Milky Way galaxy, which is just one of countless galaxies in the universe.
Solar System: The solar system refers specifically to our own star system, which includes the Sun, eight planets (including Earth), their moons, asteroids, comets, and other smaller celestial bodies. The solar system is a relatively small part of the Milky Way galaxy and is located in one of its spiral arms.
In summary, the Milky Way is the name of our galaxy, while the solar system refers to our star system, which is located within the Milky Way.