How are the differences (sound) in an instrument like a trumpet produced?

As much as I know a musicians signature sound is defined a lot by their choice of instrument, if I heard Miles playing Chett’s trumpet I would know it was Miles playing just by hearing it.

@Vignette -Are you sure? Maybe he would’ve adjusted his embouchure to imitate him more accurately.

Identifying two different players on the same horn has to do with style, not specific sound.
To tune a horn, there are passages in each of the three valve’s individual tubing that can be adjusted to get a specific sound through that valve to be on key. Otherwise the entire length of a horn’s tubing creates the specific sound, (like the difference in quality of sound between a trumpet and a cornet,) as does the bell of the horn. How large the mouthpiece is has a lot to do with sound quality, as the difference between a mouthpiece for a trumpet (small) and one for a tuba (large.)
I would guess that if the two musicians you mention played the same horn with two different mouthpieces, one player would be off key.

@lucillelucillelucille POSITIVE!! You can emulate playing style and sound but you cannot duplicate what a true master spent their lives perfecting. Hundreds if not thousands of guitarists have made a living re-creating the sounds of Stevie Ray Vaughn but as a guitarist myself I have yet to hear anyone of these guitarists play close enough to how Stevie played to where I would not be able to know it wasn’t Stevie playing.

@kritiper -TY for the answer.
@Vignette – I wonder….

@lucillelucillelucille If I played a recording of someone playing a Dollar Store trumpet and the musician was either Miles or Chet, would you be able to tell which one was playing the toy trumpet?

@Vignette – I want to say possibly but * just don’t know!*