What does it mean to be a jazz saxophone

The saxophone itself is a jazz instrument.

What you mean by "jazz saxophone" is a genre of music in which the saxophone is used to play a jazz style (the saxophone can also play pop, classical, etc.).

The jazz saxophone (style) has the following characteristics:

Includes improvisation

Uses blues scales

Complex jazz rhythms

Accompanied by unique jazz chords

Unique use of timbres such as glissandos, growls, etc.

This is the most common style of jazz saxophone.