There's also some history to it. Saxophone is a relatively new invention, dating only to the mid-19th century, well after what people think of as "classical" was mostly done. By the time classical music could adapt, it had already moved in other directions. Saxophone was adopted readily by marching bands because it's portable and loud.
Historically, "jazz" started in the 1890s in New Orleans, when black performers were first allowed to play music in public. They played what was available to them - marching band instruments. Saxophones, brass, bass drums and snare drums. String bass and piano were added as the music moved indoors. Banjo dominated the string section until the electric guitar was invented, as guitars aren't loud enough to compete with all the horns and drums.
Historically, "jazz" started in the 1890s in New Orleans, when black performers were first allowed to play music in public. They played what was available to them - marching band instruments. Saxophones, brass, bass drums and snare drums. String bass and piano were added as the music moved indoors. Banjo dominated the string section until the electric guitar was invented, as guitars aren't loud enough to compete with all the horns and drums.