I would start her on Benadryl - talk with your ped first, to get the right dosage based on her weight. Benadryl is important to STOP the allergic reaction and stop her body from creating more histamines...this is how the body "reacts" to an allergen. Histamines build in the body making the reaction, potentially worse as time goes on. So it's important to stop the reaction with Benadryl ASAP.
If you suspect it's the dog, keep them away from each other. Specifically no touching, licking, etc. It sounds a little strange that you've had the dog for months and this is just starting tho. But maybe it's the saliva. Vacuum up the main areas your child will be, just in case it's the dander from the dog too. Keep the dog out of those areas for now. Open the windows and see if you can "air out" your house right now.
Anytime your daughter gets the hives or comes in contact with the dog, I'd get her in the shower or tub immediately. Warm, not hot water. Hot water can aggravate the itching. Pat dry, don't rub.
It does sound like some type of systemic allergic reaction. A simple contact rash would (I believe) be localized to the area of contact - if the dog licked her face, then you'd see a streak of hives where the saliva touched her.
Just know that if she's breaking out like this, the next allergic reaction can become worse that it first presents. She CAN develop itchy, watery eyes, nose, ears, breathing problems - coughing, raspy voice, itchy throat, and swelling.
Since you don't know what's causing it eliminate ANYTHING new to her right now. If you think it's contact with the dog, start there. Start the Benadryl after talking with your ped. And track ALL her foods. If you notice her getting hives or them getting worse, track backwards to see what she's eaten, touched, or whatever "just" happened.
It could be food, the dog, environmental, etc. It's a process of elimination at this point.
If I were you, I'd make an appt with a pediatric allergist or see your ped soon. They can do a simple blood test or in office "scratch test" to see possibly what she's allergic to. At least then you'd "know" if it's the dog or milk or something else. For me, the not knowing is the hardest. If it's a simple test, it's worth finding out now.