It's true that dogs have an alpha for their pack - and some will prefer one person over all others - but a well trained dog obeys all the humans in the pack. I've only taken one dog for training classes, but all my dogs are trained now. The trainer made it clear that she was training us, the humans, as much as the dogs in class. We learned how to respond to the dog's behavior and most importantly for everyone in the family to respond the same way and be consistent.
It could be that the lady in this isn't following through with the training - making the dog sit before being fed or petted for example - because she thinks she being nice to the dog. Some people try to treat their dog too human like and it's confusing for the dog. I would say that probably this isn't a case of dog that can't be trained, but instead is person that couldn't/wouldn't follow through with what they were taught.
