If he hasn't yet, he should get a patch allergy test done, they do that on the arm or back and not just a blood test. Those hives and/or swellings are too close to his throat.

This looks like his body is reacting to something. It can be anything from a food he eats, perfumes or scents from products, chemicals he comes into contact with, detergent his clothes are washed in.
I suggest starting with keeping a daily journal of what he eats to see if there is a pattern when he gets the hives that can be connected to the foods he has consumed.
I also suggest going to another allergist who will do more. It appears he’s reacting to something and it’s going to take being like a detective to figure it out.

Very much like what I had and dx as Urticarial Vasculitis.