When I play a note on my synthesizer or my guitar, it is exactly one frequency plus the overtones and subfrequencies which are doubles and halves of the tone.
As per Tetsujin's comment, that's not always the case; there will often be inharmonic partials, variations in the frequency of individual partials, and regions of energy that perhaps don't have a distinct frequency. But you're right it is also often possible to create sounds that stay close to perfectly periodic (and therefore have only harmonic overtones) at least for a short time.
when I listen to a vocalist, it sounds like the vocals are not just "one frequency" but that the voice has many different frequencies at the same time.
The human voice also tends to be an unstable sound - the singer may be deliberately varying the air pressure, pitch, resonances of the vocal tract and mouth (which will emphasize different frequencies within the note) - or these may be varying involuntarily. As well as the basic frequency of the sung note, there may be a periodic vibrato taking place at a different (much lower, non-'audio') frequency, and if the vocalist is making a growling sound, you may hear the aryepiglottic folds vibrating at yet another frequency. Additionally, there is a lot of turbulence and 'noise' in a human vocal sound.
However, though the human voice is complex, there isn't any essential characteristic of it that can't be found in other instruments or sounds; I don't think you're hearing the human voice wrong, but perhaps you haven't noticed the subtle complexities in other sounds yet.