Bass, of the two you suggested, Excel sounds likeliest. In photos, the leaf looks pretty variable and certainly covers the bases as far as how the leaves look on mine; also on the fruit I photographed, the eye was closed and the skin was very delicate but the info I read about Kadota said it has tough skin and a drop of "honey" in an otherwise-open eye. I got cuttings of both Excel and Kadota from UCD, so assuming they root and thrive, I'll be able to compare them side-by-side.
Of course, my assumption that it's a common fig from a local nursery has nothing to back it up but guesswork. As you say, "it could be anything." I'm in an old neighborhood (my house was built in the 50's) and there are a couple of other fig trees (purple) on my block that the owners don't know the varieties. I assume both are black missions, but figured I'd check them out once they have ripe fruit to see if they look/taste like mine. Even so, they could be anything as well--I just don't have enough experience to make a very informed assessment.