Can a child with allergies to dogs and cats become immune to ones they grew up with?

Not really. The allergic reaction to pets is caused by the dander the pets have. It may be that your pets are cleaner, given the fact you are aware of your child's condition, you may be more vigilant at keeping the pets cleaner...therefore, they may not have as much dander. If your child's asthma is triggered by pets (which is the number one allergic trigger for asthmatic reactions), then all pets would cause this. Asthma can also be magnified by increased activity and physical and emotional activity. It may be the child is more active or gets more excited out of the home in different environments. Consider many factors. Check the child's activity level, emotional level, foods, air quality, medication compliance etc. With the right diet, activity level, moderation's and medications asthma can be controlled. Please follow up with routine MD exams and comply with the MD regimen. Best of health to your child.

your animals are probably cleaner animals, the dander is the allergen, if you wash the animals in an allergen reducing shampoo this will help!

A recent study has shown that encountering cats in early life triggers asthma in susceptible individuals (i.e. those with a genetic predisposition) but that encountering dogs may actually protect against it. Keep your daughter away from cats.