If this were a man instead of the lesbian, there would be absolutely no question even among liberals that he had no legal right to this child.
The rest of that statement was unnecessary. Liberal-dominated family courts only care about fathers when it comes time to pay child support (and, of course, when it comes to child support not being the biological father is no defense if a support order has already been issued). Father visitatation is used only as a weapon against the father - it is not viewed to be a good thing in and of itself.
As for the father having sole or even primary custody - dream on. The number of fathers for whom this happens is statistically insignificant.