No, that is not part of the MIDI file specification.
Regarding the question, the midi file contains refrences to specific instruments (and percussion) as defined by the General Midi specification and the instrument (in this case, your computer using a sound font) decides what it will sound like.
If I plug MIDI into my Roland, the instruments sound very different than if I plug them into a Yamaha even though "piano" is "piano" and "flute" is "flute".
Midi files contain a whole lot more data than that (actual notes, duration, inflection, etc) but it is not related to your question.
MIDI files weren't invented for computers to play/render them (as in produce sound). MIDI files were made for recording and controlling midi instruments and effects FROM a computer.
As a side note, in theory, a SysEx message could have technically contained the information you wonder about but I assure you that it isn't there as nothing else would understand what to do with it.
Some of the most fun to be had here is trying different sounds with the parts :)