I think that shorter strings have a more pure sounding note. This makes sense (to me at least) as the longer the string the more modes it will vibrate in for any given input. which, in turn, equates more harmonics. So as you play the same phrase further up the neck the timbre definitely changes. All things being equal I will play a phrase in the position that has tonality that suits the piece.
(checked with my daughter who plays violin. When playing in an orchestra they all have to play the notes in the same way)
Take a skipping rope get someone to hold the other end. You can vibrate one end and get it to oscillate at the fundamental, third and fifth harmonic quite easily. Now try again from the middle you will find you need much more energy to get harmonics with a shorter string.
On an instrument the impulse or bow will be about the same when the string is fretted so the same energy applied to the string will result in fewer harmonics.