I'm going to try to respond in detail in case someone without your depth of knowledge is interested. You may have to skip some things.
I do not understand this exactly, "If you remove the antibodies binding Galactose-alpha-1,3-galactose from the sera, you remove a lot of the antibodies that react with dog dander."
This is the common method to identify cross-reacting antibodies. A person makes a wide variety of antibodies to several antigens in the tick saliva, here the interesting one is alpha gal. A simple way to show that the alpha gal antibodies induced by tick saliva are the same as antibodies to dog dander is by absorption. You stick the alpha gal carbohydrate on something solid like a tiny bead and expose the the person's serum to the bead. The antibodies that stick to alpha gal then stick to the bead. You then take the rest of the serum that isn't stuck to the bead and test it for binding dog dander. In this case, there were fewer antibodies binding to dog dander indicating that some of them stuck to alpha gal or cross reacted with alpha gal. Since dogs can make alpha gal and add it to different proteins, the simplest explanation is that part of the reaction to dog dander is to alpha gal.
There is a link between cat and pork allergies, maybe something similar - I am terribly allergic to cats, and came up allergic to pork, and lamb. Also allergic to dogs.
(I won't tell Jazz) Well it could be a reaction to alpha gal or it could be a reaction to any of many antigens common to those animals.
So, indy was allergic to cats, but not to dogs, according to her allergy test. Might that mean then that she was not allergic to this substance?
Indy is a different story because she shouldn't react to antigens that are common to dogs as a species and alpha gal, I believe, is common to the whole species but not primates. So probably Indy was not reacting against alpha gal but something else in most dogs but not her.
Might it mean that I am since I am allergic to both? Or are there just too many possible allergens in the dander?
You may react to alpha gal but there's not enough information to be sure.
So basically, if both people and dogs get exposed to alpha gal in tick saliva, I don't think that dogs will react as violently to meat as people, because dogs themselves make alpha gal and ought to be tolerant.