Just picked a couple Black Brazil figs. They are most likely the variety known in Brazil as Roxo de Valinhos.
"Although there are about 25 varieties of tree in Brazil, only one is commercial. This cultivar, called 'Purple de Valinhos', was introduced into Brazil by an Italian immigrant who cultivated the tree in Valinhos ('Roxo de Valinhos'). It is a rustic variety, vigorous and productive, with good adaptation to different climates that occur in Brazil, besides being adapted to the system of pruning. The fruit is elongated, large and Pear, with short stalk, skin color and dark purple in the inner flesh, pink-violet. The fruits can be designed both for the fresh market "and for industrialization, in the form of sweet syrup and crystallized.

So are there common features--whether in terms of hardiness, productivity, disease resistance, taste, or whatever--for figs from one geographic region as opposed to another? E.g., do figs from say India share a set of traits that differs from most Italian or Turkish or Greek figs?

They're all the same sub species of ficus. Some varieties were selected for their performance in certain climates. The same fig might be productive and tasty in one area and not the other. In the states we probably have all the climates of the world, so one variety that comes from a humid country might do well in Florida.

I hope that answers the question.