Experience has taught me that I do not have a singing voice. But in practice, I tend to sing from my lungs and speak from my mouth.
What is that person trying to say?
How can it be possible that a person does not have a singing voice?
I think he may mean that he doesn't identify himself as having a 'singing voice' that is separate to his 'speaking voice,' but that he's noticed that when he sings, he supports the sound with a more controlled airflow (he says lungs, but in practice this is probably support from the diaphragm while keeping his ribs wide), and when he speaks, he doesn't support the sound particularly so it feels to him like the sound-making is coming just from the mouth.
This is close to how I feel it too. I speak within a particular area of my range, and tend not to support the sound well. When I'm singing, I can go higher and lower than my speaking range, through the help of various muscles. Janice L Chapman in her book 'Singing and Teaching Singing' explains how various muscles are active in different registers.
There seem to be differing opinions on singing voice v. speaking voice. Here's an interesting article focussing on the difference between speaking and classical singing (focussing on larynx position): http://www.britishvoiceassociation.org.uk/articles_singers-speaking-voice_speed.htm