An animal will take up more space than the person would on their own. The animal would be a passenger. I do not believe airlines should charge full price for a service animal, but there should be a charge. I do not feel that would be a discrimination against the person or their animal.
In an airplane an assist dog is just like any extra, they take up space, add weight and assist dogs probably require other considerations for them to travel safely such as special harnesses. Airlines are already under a lot of pressure to keep expenses low, and an assist dog is not a low maintenance item. If something would happen to the animal in transit it would not be as easy for the airline to compensate as it would for a piece of baggage. People who need assistance shouldn't consider themselves a special case, but should gladly pay a fee for this kind of special consideration.
Charging an extra fee to those requiring assistance dogs would be discriminating against the disabled. If someone requires a dog for daily living, then that dog should be welcomed on the flight, and should not cost extra. We don't charge extra for those who need a wheelchair, or crutches, and those things take up extra space as well.
Airlines already charge people for any and everything while at the airport. I do not think they should charge people who need assistance from dogs. Airlines should only charge if people bring a dog and do not need assistance from them. It would be a choice in that case. People who have assistance dogs need them and should not be charged because of it.