The films posits a hypothetical future world government whose primary tyrannical element is that any male female relationship both people must be screened for genetic relationships and if related them pursuing a relationship or reproducing is illegal. IVF and other reproductive tech proliferate and apparently the citizenry get freaked out by the idea of accidental incest. A lot of the plot is also spend on some kind of travel visa, but that is a feature of the world already. Needless to say our hero runs into an issue.
Ok I admit it is a horrible violation of rights in principal, and a giant waste of effort, but if I had to live in a future dystopia I have to admit it isn’t too bad. I mean hell wouldn’t it be more realistic to just map everyone at birth and create a library of genetic info that doubles as a bizarre matchmaking service?
Been a while since I saw it but I think the OP misunderstood something. The screens weren’t about relatedness, they were about the ability to reproduce without risking having children with any conditions requiring treatment. The travel visas were closely related to this because they were related to insurability–and so were the screenings. Outside city walls you could be free to have any defects you wish–and you were uninsurable, which those inside thought of as the kiss of death. Indeed those found uninsurable were iirc exiled to the outside.
The Wiki summary suggests it’s relatedness but that genetic or other biological issues (one person illegally travels somewhere he’d been denied entry to based on screening, then dies there of a virus he had no resistance to) play a part as well.