Change of name in birth certificate

My son is 4 year old . In his aadhar card her mother’s name is mentioned not the father name like this : Chitraksh girdhar s/o Neha chugh I want to ask is it mandatory to enter father's name in kids aadhar card or mother's name is also fine ? 2nd query : In my son birth certificate mother's name written as ’Neha ' but in all other documents her full name written Neha Chugh ..i want to ask whether she need to give her full name in birth certificate or first name is also fine ? I hope u understand what i am asking ? Please reply Thanks ..