It is not type-specific, because it is not treating a type. It is putting back the fats a barrier has lost, and that is the same job whatever the eczema is called.
People use it on atopic dermatitis, contact dermatitis, dyshidrotic, nummular, and on thickened leathery skin from years of scratching the same spot. Hands, knuckles, wrists, the creases of elbows and knees, the neck, the face. It does not sting on cracked or broken skin, so it can go on the worst of it.
What it will not do is make eczema go away. Eczema is chronic and it relapses, and anyone telling you otherwise wants your money. If your child is on a prescription, keep using it and tell your doctor you have added this alongside.