Is it permissible for a married woman to adopt a child from a girl who intended to get rid of it, and attribute it to herself, and is this child considered her legitimate son according to Islamic law?

Saving the child's life is a righteous deed for which you will be rewarded. There is no harm in him calling you "my mother" out of respect, but he should be attributed to his legal father, not to your husband. The child is not one of your mahrams unless he was breastfed by you or by one of your daughters five full breastfeeds. The child's mother should be advised to repent from her intention to abandon him, and to repent from her heinous act if the child is illegitimate.