Its made of deer hide backed with some type of hard canvas or something on the interior.
First off, saddle soap is a cleaner, not a softener. All you're doing is making a very clean stiff bag. The only thing softening it is the working of the leather when you apply and rub out the soap (see 2 below).
There might be a stiffener between the canvas and leather. If there's a non-obtrusive spot on the inside you can peel a little of the lining to see what's really going on in there... I'd look at that first. If there's something besides just canvas and leather, you might never be able to soften it up much. If not, here's a couple of tricks I use that work...
1- Neetsfoot oil. Might darken your leather, but will soften it very well. Get a good brand like Fiebings, watch out for the over-priced artificial junk they sell in some of the shoe stores. Apply, rub out, repeat.
2- Toss bag and ONE old canvas skeaker into a dryer, put it on NO HEAT and run for 10 minutes. The sneaker will beat the bag, working it to soften the leather. After the first application of abuse, if it's not soft enough, go again for 5 minutes. Repeat running the dryer for 5 minutes each time until you get it where you want it.
IMPORTANT: NO HEAT Heat will toughen and dry the leather making it worse.
Hope this helps,
vic