Overtones are created as a function of the fundamental frequency. That is, fundamental frequencies and overtones are related to the vocal folds. Vowel is not determined by the vocal folds. Vowel is determined by the first two vocal tract resonances (also sometimes referred to as formants). The space from your vocal folds to the place of maximal constriction created by the tongue determines the first resonance. The space from the place of maximal constriction created by the tongue to the lips determines the second resonance. When frequencies pass though these spaces, the size and shape of the vocal tract will have certain frequencies that resonate within the cavities. The combination of these two resonances (regardless of pitch) will reach the ear and be interpreted by the brain as a different vowel quality. Each vowel has a certain range of resonant frequencies that will be interpreted as said vowel. This is individual for each person depending on a lot of biological factors.
What is interesting is that in singing it is possible to sing above the range of the first formant (depending on vowel) when this happens a singer can lower or raise the first resonance (depending on style of singing) to track or detune the resonance which will give a different timbral effect. However doing this can also change the quality of the vowel. This is called vowel modification and has been used by singing teachers for centuries.
TLDR: Overtones are affected by vowel space but are not determined by vowel space. Vowel spaces are different for everyone and are interpreted by the brain separately from overtones.