It's the other way around, actually. Our Solar System is a simple planetary system made up of a star, planets, dwarf planets, moons, asteroids, meteoroids, comets, and cosmic dust. The Milky Way is a galaxy made up of hundreds of billions of stars, stellar remnants (neutron stars, black holes, etc.), cosmic dust, and dark matter. It's unknown how many of those stars have planetary systems, but 54 (including our own) planetary systems have been confirmed, and it's estimated that the Milky Way galaxy could have as many planets as stars.