Considering you can do it online with no hassles - sounds like the reporter garbled the story. Maybe some other tests they do (not paternity) require a prescription because they would be giving medical advice/information?
NY State requires a prescription or a court order for a paternity test. (cite) The reason is to establish a chain of custody so that the results can be admissable in court.
The FAQs for the online services all point this out. The over-the-counter tests you can buy are different in NY than elsewhere.
They’ll sell you the kit at a drugstore or online, but you won’t get results without the prescription.
Most DNA labs, especially the on-line ones - if you want a DNA test with chain of custody, full legal rigamarole and that it stands up in court - yes, it has to be done by a medical professional and they need to take care that they get appropriate ID so nobody is pulling a fast one; they handle, label and certify the swab so that nobody can substitute on the way to the lab, too.
If you just want to swab Junior’s cheek while mommy is not looking, then mail his and your swabs in for a private assesment, just to - as Ronald Reagan would say - “trust but verify” then you end up with a result that will not stand up in court but might tell you if mommy has been doubling down.
So presumably then, this service is designed to provide a legal test - I assume because it’s a legal requirement and the people of that area have trouble finding another venue for the test? Some states require welfare mothers to identify paternity; at which point, the (non-)father can get out of support obligations only through a court-valid DNA test. Either way, a valid community service.
This is a really stupid reason. There are reasons people may want to get a paternity test done that have nothing to do with legal matters.