Probably because of you don't take proper precautions an omega or alpha can start a riot or mating frenzy if they leak pheromones, the government in this stores likes to monitor these potentially risky people. That's still wildly horrible to think of people being tracked and monitored because they were born with certain genes though. And the hyung avoided his little brother being officially classified as a government recognized omega because even in real life discrimination gets way worse once there's a paper trail announcing to everyone that you're a disliked minority.
Bec people can exploit you. Especially omegas.
I don't understand why it's illegal if you don't register as an alpha or omega. That makes no sense. I think they should've explained the reasoning behind that. It's not like they're espers or something.