Being allergic to a puppy would mean that you would probably be allergic to the puppy as it turns into a dog as well.
So you could have a puppy and another dog and only be allergic to the puppy and not the dog but if you're allergic to the puppy you'll usually be just as allergic to the puppy when it grows up into a full grown dog as you're to the puppy right now.
Most times the allergies to dogs are caused by the dander that the pet puts into the air and not the dog itself.
Sometimes brushing the dog and grooming the dog often enough and properly will help with the dander and pet allergies.
However sometimes you may have to give the dog or puppy up due to you or your child's allergies.