I'm a American music student and I'm similarly confused by your question. I very frequently hear voice referred to as a musical instrument. It requires skill and practice to use, and produces music.

I'm not sure I really understand your question. A musical instrument is, by definition, an instrument used to generate sound for the purpose of music. The human vocal system is well suited to this. Do you mean somehow a view of the human voice as an 'external' instrument? I'm confused by your question.

In De Institutione Musica, Boethius (480–524) classifies music in musica mundana (planetary patterns, the zodiac), musica humana (fevers, rythms of the human pulse, breathing, changes in the body), and musica instrumentis. The last one would be what we call music, that which is produced and is audible here on earth by us humans. Musica instrumentis would include music produced with whatever instruments and/or the voice.

Those classifications can be found for many centuries after him, this book was read by medieval scholars (studying music at a university had nothing to do with actually playing/composing music, it was philosophy). So yes, in that sense the voice was an instrument.

In practice, vocal music was distinguished from instrumental music. For many centuries, vocal music was the prominent way of making music (instrumental music didn't started to really take over in status until the 18th century).

Did people think they had a tiny instrument inside? Aristotle had already knowledge of the larynx, Claudius Galen also studied it and identified it as the source of the voice (see this paper and also this one). I cannot tell you what the average person thought, but at least we know Antiquity manage to pass that knowledge (I don't know how widely read those texts were in the period of your question).