I finally have ripe fruit from both trees to compare.
In summary, I have 2 trees growing in my yard that look very similar. One is Hardy Chicago and one is an unknown I've been calling Joe's Jersey.
The Hardy Chicago was started from a sucker taken from this tree in Methuen, MA in 2010 (the original source for this HC tree was Raintree Nursery).

The Joe's Jersey was started from a cutting from this unknown tree in Clifton, NJ in 2010. The tree has been at this location since at least the early 1980's. It has been severely pruned back several times but keeps re-sprouting new growth.

Both were both planted in my yard in 2011 in early spring. Here is Hardy Chicago today.

Here is Joe's Jersey today.

The Joe's Jersey tree has been ripening fruit for a couple of weeks. Hardy Chicago is just starting to ripen figs now. Here is a comparison of the fruit from both trees today.


They taste the same. Both are very good. The desert heat brings out a nice depth of flavor in the fruit.
If I had to make a guess I would say they are the same variety, though I think that the Jersey tree is a little bit healthier. It grows better and makes more fruit. It might just be a clean healthy strain of HC. Either way, I'm happy to have both growing in my yard.