Do children whose parents were never married have the same legal rights as children born of a marriage?

Yes.  Children whose parents have never married have the same legal rights as children born to married parents.  RCW 26.26.106.  An increasing number of children in the state of Washington., as elsewhere, are born to or adopted by non-married parents.  When unmarried parents’ relationships fail, the needs of these children are the same as children of married parents.  The law is based on the premise that all children need to be supported by both parents and to continue nurturing relationships with both parents, if neither parent  is impaired. If a parent is "impaired", for instance, by uncontrolled alcoholism drug addiction, the child's relationship with that parent will be limited by the court as necessary to protect the child's safety and welfare.   If both parents are willing and able to provide adequate residential care, both parents will be entitled to a share of the child's residential time.  The proportion of time the child spends with each parent will be decided by the parents, or by the court if the parents cannot agree on a "Parenting Plan".  Plans providing for short, substantially equal periods of time with each parent are available if the parents agree to such Plan, or if the court determines it is in the child's best interests even if the parents do not agree.  RCW 26.09.187.  The standards the court applies to resolve Parenting Plan disputes between unmarried parents are the same as those applied to children of married parents.  

Does a biological father have the same rights and responsibilities as a married father? 

Unless both parents have "acknowledged" the paternity of the child, and the father is consequently listed as such on the child's birth certificate, the biological father has no residential rights or support responsibility. 

Unmarried biological fathers who have “acknowledged” their paternity of the child have rights equivalent to divorcing fathers, to have time with the child and to participate in parental decision making.  Biological fathers whose paternity has been adjudicated by the court, also have rights equivalent to divorcing fathers.  RCW 26.26.101. 

If the biological father is not “acknowledged” either parent or a guardian on behalf of the child can seek to have the father’s identity determined by a “parentage” (formerly known as “paternity”) action.  RCW 26.26.500.  In some situations, the state of Washington may initiate a legal action to establish parentage and impose support, at the request of either parent.   If the mother seeks certain state aid on behalf of the child, the state of Washington will seek to establish paternity in order to be able to reimburse the state for benefits paid on behalf of the child.  RCW 74.20.