The sound-producing mechanisms in string instruments and the flute are entirely different. Are the pitch-producing mechanisms used while singing a song and humming a song (humming with closed lips and no tongue movement) the same, or entirely different?
You can try this for yourself. Sing a note, to any vowel sound (try them all) and whilst doing so, close your lips. Now you're humming that same note. Feel any difference? Probably not, because the note itself is made further back. Somewhat like putting a mute on a trumpet bell, or covering a loudspeaker with something. All you're doing is muting the sound by closing the lips and humming.
So, for Stinkfoot in particular - the answer to 'Is the pitch-producing mechanism for singing and humming the same or entirely different?' is actually 'No' , with the caveat that it's pretty damned close rather than entirely different, at least as far as the layman's concerned. Go (a lot deeper) and it's not exactly the same, but for all intents and purposes says 'yes'.
As Tim suggested, here is some additional information. His answer, by the way, is (sorry in advance) pitch-perfect.
While the fundamental pitch depends only on the vocal cords, the overtones and whatnot depend on the complete shape of the acoustic cavity, the vocal cord waveform (which tends to be amplitude-dependent), and so on. To clarify a little- the throat, mouth, and nasal cavities will produce various harmonics as well as phase-shifted copies of the original sound. The vocal cords, I believe, do not vibrate as perfect sine waves, and further the waveform will change with amplitude, thus changing the overtone series at the origin.