I don't know who is the step-parent here, most likely your partner, but here's the thing. In all states, the biological parents have superior rights to anyone else. That includes grandparents, siblings, cousins, step-parents, etc. The biological parents have the rights to be with the child and nobody else can have custody of the child unless the parent is deemed to be unfit or has neglected or abused the child.
The step-parent, however, if living with you and the child, can do his very best to fill in because the kids need a father figure. If they don't have a relationship or much of a relationship with their father, there is nothing saying they can't start forming one with their step father. My daughter is very close with her step father because her father was on the other side of the country. She has remained very close with him and now she's an adult.
So yes, the step-parent can try to fill in where the custodial parent is not in the kids' lives enough, but he can't bring a lawsuit for custody. That can't happen unless bio dad is unfit, and there 's nothing in here to make me think that the dad is unfit, unless there are more allegations that I don't know about. Feel free to tell me if there are other issues here.
If at any point the bio dad disappears off the face of the earth, then you can talk about abandonment and adoption, etc. but at this point, if dad is still around, the step-parent can try to fill the dad's shoes at home.
Does this answer your question?