No, this is not the case, and the court does not use the term custody.
The court can make orders determining who children should live with, which may be the child living with both parties on a shared care basis, or for the child to live with one parent and spend time with the other.
The mother is not given preference by simply being the mother, and the court will determine these issues based upon what is in a child’s best interests, and based on the welfare checklist detailed above.