Have you ever heard of the trick to cook a potato faster by pushing a nail into the center of it before placing it in the oven? The nail gets hot and so the potato gets cooked from the inside while also being cooked from the outside..
I've never heard of using stones in soup, but it immediately reminded me of the nail trick so my reaction was "ohh, that's a really good idea." Lol.
Some stones (maybe all stones?) retain heat very very well. In fact, I once ate at a restaurant called"The Stone Grill" who's signature dish was bringing you a hot flat stone slab along with raw steak \ chicken \ pork and you would cook your own meat on the rock. I was full after I finished eating all the steak and pork so I threw my raw chicken on the stone to cook so I could take it home, and even after 40 minutes the stone was hot enough to cook the chicken..
Soo yes, old wives tales and stories aside, stones cooked in soup would most certainly keep the soup hot long after it was removed from the stove. And might even help it cook slightly faster..