I have a baby fig tree growing in my back yard; this is unexpected! Not sure where it came from. We do have two other fig trees in the yard. One is several years old, about twenty feet away. The other is only two years old, also about twenty feet away in the other direction. Both lost their leaves several weeks ago. Both are unknown cultivars. This baby fig is only now losing its leaves. Do figs send out underground runners and send up suckers twenty feet away? Or do new trees sprout from a fruit that may have been dropped there by a squirrel? In any case, I can't keep it, so this Spring it will probably be dug up, put in a pot, and given to somebody who can give it a permanent home. (Anybody here interested?)
This is most possibly seedling from dropped fruit. It is highly unlikely, that this is a sprout from root of another tree. Only if your other figs are very old and have enormous root system, then there is a slight canche that it's a coppice sprout from damaged root. Fig seedlings are likely of smyrna type and need a wasp. Their properties are likely not the same as was on their parents. It takes several years before there is a hope to see, if figlets stay on or will drop without pollination.
Your best bet to grow a fig tree is to root a cutting from a known cultivar. The chances that a seedling of 2 unknown cultivars would be anything special is remote . . . but, imagine if it were something new and wonderful!? It's the stuff dreams are made of. If you have the space, time and inclination to grow your chance seedling to maturity, it would be fun to discover its unique characteristics. If it should turn out to be something great, you could name it whatever you like and wait for the royalties to roll in.
