If the label says Dexron III, you will be getting Dexron III. It is still sold by many different oil manufacturers.
I've been using many brands of it over the years (including since their ending the licensing in 2006) and have never bought or used a bottle of ATF made by GM, ever.
Obviously GM won't recommend it (as there is no profit in it for them), and you can do as you please. But as long as it does no harm, I'll use it.