One would assume that we only have one mouth, and I guess that is technically true. When we think of the mouth we are thinking of where our tongue lies, and many might even think this is where we form vowels.
However, for a singer the real mouth is in the oro-pharnyx (oro means mouth). Basically, the area at the level of mouth but in the pharynx, or what you might think of as the space right above the throat which houses the larynx.
What happens in vowel formation is that the mind thinks of the vowel and that information is transmitted to the vocal cords. The vocal cords shape themselves for the specific vowel as well as pitch. The cords send this information through the now vibrating air molecules to the oro-pharynx area which is shaped accordingly, and the vowel is pretty well formed.
[Of course, if there is a higher pitch then the air travelling process is through the oro-pharynx to the sinuses then returning to the oro-pharynx where the vowel is then pretty well formed. Remember that this process involves pitch and is not just concerned with the vowel.]
The “shaped” and vibrating air travels into the mouth area where finishing touches are put on. The air molecules then travel into space where they come into contact with the human ear. The ear translates this information as a particular vowel at a particular pitch and volume.