Considering the vocal cords can be a membranophone or considering the air column can be an aerophone. or do we need another classification?
I'm not sure this is really an answer, but I thought I should elaborate on my comment which may have seemed glib or snarky.
The classification list is the Mahillon and Hornbostel–Sachs system which I really didn't know about.
I'm not really sure about the logic of that system. It lumps together bells and maracas as idiophones, because the entire bodies of the instruments vibrate. I get the idea, but don't see any musical insight from it. Then again this system is 19th century "science" ...that's the same time period of phrenology. I think it's a little outdated.
Apparently there is a Dewey Decimal-like classification in that system. An aerophone has a vibrating air column, but the instrument itself doesn't vibrate (not sure that's true, especially brass) so that would not be the human voice as the vocal chord vibrate. Seems the voice is a "242, Singing membrane, vessel kazoo," because it's a vibrating membrane inside a container.
My original comment - "voice, the first instrument" - was meant say I think we should sometimes dispense with academic systems. Surely, the human voice was the first musical "instrument", the first means to express emotion with sound. I don't really care that it's like a kazoo. I care more about the voice's intrinsic qualities - the fact that it's easy for most people to make a distinct pitch, typical range of the voice, amplitude, and the limit of what can be sung in one breath. I think many musical fundamentals stem from what we can do with our human voices. IMO, it's more important to know the musical implications of our human voice limitations rather than classification as a vibrating kazoo-like membrane in a box.