Thanks for the photos. These are helpful:
This is very likely to be chronic hives or urticaria. The pet can be causative, as you mentioned.
The major causes of urticaria are;
1) Foods or food additives
2) Inhalation, ingestion of, or contact with antigens
3) Drug reactions
4) Infections: bacterial, fungal, viral, and helminthic/parasites
5) Insects (papular urticaria)
6) Cold urticaria
7) Aquagenic (water) urticaria
8) Solar (Sun) urticaria
9) Chronic autoimmune (problems of immunity) urticaria
10) Chronic idiopathic (unknown reason) urticaria
So it is a sort of detective work to find the cause.
Oral anti-histaminics are effective and you may try Benadryl.
A chronic urticaria is treated as following (may have been done for you by your dermatologist);
1) A consultation with an allergist, if not yet done.
2) Avoid foods with known food allergies.
3) Avoidance of mental stress over tiredness, alcohol, nonsteroidal anti-inflammatory drugs, and tight-fitting garments.
4) Psychologic stress can trigger or increase itching, so taking care of those factors.
5) Night time symptoms are helped by lukewarm bathing and keeping the temperature of the bedroom slightly low.
6) Itching is significantly helped by menthol and phenol cream.
7) Non sedating antihistamines. The dose may need to be doubled or quadrupled.
8) Hydroxyzine or doxepin in night time.
9) Colchicine or dapsone is given who do not respond to above.
10) Autoimmune urticaria should be investigated and one may benefit from methotrexate or cyclosporine.
Please feel free for your follow up questions.
I would be happy to assist you further, if you need any more information.