That's a good point, you can actually change the way a sax sounds quite strongly, there are even people who modify their reeds. The difference between the way the same instrument family sounds on say Pink Floyd's "Dark side of the moon" (Dick Parry) sounds versus say Brandford Marsalis, Michael Brecker, Fela Kuti, Grover Washington or (god forbid) Kenny G is huge.

Of course 'saxophone' is a whole range of instruments going from Low-E giants to saxes so small you have to have pretty small fingers to not hit two valves at the same time.

The mouthpiece and the reed thickness are a big factor in this as is air pressure, 'tonguing' the reed and how you shape your mouth cavity. Endless variation.

But the basic harmonics are baked into the instrument, those would be very hard to change though I'm sure it is possible to manipulate which ones get amplified by the above changes to the environment.

https://www.youtube.com/watch?v=kZjZBe6o78M

Is a nice example, it would be a pretty boring song without that raging saxophone.