What common charactaristic do all the woodwind instruments share that makes them all in the same catagory?
As a woodwind player and arranger myself, they actually fall into two categories, reed instruments (which saxophones also fit into) and hole instruments (like the flute). These are both woodwind categories, and the wind category also includes brass. It can also be defined as blown instruments where the instrument produces the vibration – in brass, your lips provide the vibration, and in woodwind either the reed or the shape of the lip-hole provides it.