According to the article, the couple want to keep and bring up their child, but they are worried about the possibility of the biological progenitor(s) turning up and demanding custody. This is a realistic fear.
I know nothing about clinics like this — I wonder if they keep a dna database of clients? that would be easy enough to check. But I have no idea if that’s required by law and, I would wonder if this particular clinic even bothered.
The other side of the coin, that I don’t see mentioned in the story, is that somewhere there is a baby they conceived being raised by someone else. Or their baby was discarded.