The lifestyle of the lesbian couple is at the heart of this story, fairly obviously, but the fathers sexual orientation is neither here nor there really.
Central and fundamental to this question is whether the friend was essentially a sperm donor, or whether he expressed an interest in and acted in the role of a father.
He says he is just a sperm donor - and as far as i know, we would not expect sperm donors to assume any kind of financial/legal responsibility for any children that were the outcome of their donated sperm.
She alleges there was an agreement that he would assume a fatherly role in bringing up any offspring, and indeed for a while he fulfilled those obligations. It is only in these circumstances that he might have some sort of financial/legal obligation, as far as I can see.
Up to the judge/court to decide whether he is shirking his obligations and reneging on a promise, or whether she is a money-grabbing chancer.