Hmm...I can see the reasoning that if baby shampoo is gentle enough for baby maybe the vet thinks it's gentle enough for puppy. But I chose to stick to dog products.
Scottie initially was always itchy. I don't actually know why... whether it was his food or his shampoo.
Then, I started using Earthbath oatmeal and Aloe (fragrance free) dog shampoo
Amazon and also started feeding pea-sized amounts of organic virgin coconut oil to him daily and his skin/itchiness got better.