Our galaxy, the Milky Way, was given a name because it is a distinct structure that can be observed and studied from within. When astronomers looked up at the night sky, they noticed a hazy band of light stretching across it, which resembled spilled milk and gave rise to the name "Milky Way." This name has been used for centuries to refer to our galaxy as a whole.
On the other hand, our solar system consists of the Sun, the planets, their moons, asteroids, comets, and other celestial bodies, all gravitationally bound to the Sun. The term "solar system" is a general description of our planetary system revolving around the Sun. It doesn't have a specific name because it is not a unique structure that can be observed from a distance.
The naming conventions in astronomy often involve giving names to individual objects or specific features within a system, rather than naming the entire system itself. For example, the planets in our solar system have individual names, as do many of their moons. Similarly, prominent features within the Milky Way, such as nebulae and star clusters, have been given specific names.
In summary, our galaxy, the Milky Way, was given a name because it is a distinct structure visible in the night sky, whereas our solar system is a more generalized term referring to the collection of celestial bodies orbiting the Sun.