In a word – yes: a non-biological father (e.g., the mother’s new husband or subsequent de facto partner) can be awarded “custody” of a child if such an Order is considered to be in the child’s best interests.
A non-biological father (or, indeed, mother) can apply to the Family Court as a person concerned with the care, welfare, and development of a child, in the same way as would a grandparent or aunt.
If this type of Application is made, consideration will need to be given to the role that the non-biological parent has played in the child’s life to date; the child’s relationship with that person; and the child’s relationship with their own natural parents.
Generally speaking, if a child already spends substantial time with both his/her natural parents, a non-biological father after a subsequent separation may struggle to obtain Orders to spend time with the child, let alone achieve an Order for any form of parental responsibility.
However, if the biological father has been largely absent from the child’s life and the child has formed a very close, quasi-parental relationship with the non-biological father, the Court would be more inclined to positively consider such an Application.