Hi there, a cough is a common symptom of a pet allergy so it is very possible this is to blame for your symptoms.
The irritation is caused by animal dander (the urine, saliva and dead skin cells of animals) rather than the animal's fur. Animal dander contains proteins that are different in cats and dogs and so this may explain why your cat did not cause the same set of symptoms. The dander from male and female dogs can also differ, with some research hinting that male dogs may cause more allergic rhinitis symptoms than female dogs due to the protein their dander contains.
You can read about these topics in my blogs:
As the cough hasn't gone away and your wife has finished her course of antibiotics it is worth going back to the doctor for another consultation. Don't forget to bring up the fact that you have a new puppy (one of my favourite kinds, by the way, very cute!).
In the meantime, drink plenty of water, try to keep the dog out of the bedroom to minimise the spread of dander there and read my blog 'Fight allergies naturally' for more advice.
Do you have questions? Please feel free to ask.
Take our quick quiz to discover just how healthy your immune system is, as well as some useful information about your general health and wellbeing!