Hi
Itching doesn't transfer via milk.
Could be atopic dermatitis as the onset of problem is from 6 months of age. Both genetic and environmental factors contribute to it.
Apart from medicines, general measures are vital to treatment.
Next Steps
consult a dermatologist with photographs for treatment.
Health Tips
use cotton clothes, use mild cleansers, apply mositurizer liberally all over body on damp skin 2-3 times, use Luke warm water, short baths recommended, avoid excessive heating or sweating.