The pathogenesis behind Aquagenic Urticaria isn't definitively known - and the extreme rarity of the condition makes studying it difficult (only ~100 cases published!)
It does appear to be an allergic-type response - as shown in the linked article from your question the wheals are formed when histamine is released and AU appears to respond to antihistamine treatment in most cases.
One theory (Czarnetzki et al ) is that the patient isn't having an allergic reaction to the water itself but rather a water-soluble antigen present at the epidermal layer - and that when the antigen is dissolved in the water it then diffuses through the epidermal layer causing the mast cells to release histamine and produce the wheals.
However that is probably not the full story - since there have been reported cases where there was no signs of a histamine response and treating with antihistamines proved ineffective (Luong et al)
So to summarize - no definitive mechanism as been determined, but it's probably not a case of simply being allergic to H2O molecules.