I'm thinking a sort of dystopian inspired setting where birth rates have really plummeted and infertility is common. Omegas are forced to undergo testing once they're adults to determine if they're fertile and would be able to carry to term. If so, they're involuntarily inseminated during their next heat (maybe the heat itself also is induced through artificial means) with sperm that is a mixture of a bunch of alphas. The alphas are like, high society and from rich/powerful families, who have all undergone genetic testing to ensure they could make healthy, strong pups. Since fertile omegas are so rare, they're in high demand, so to keep things 'fair' (take the use of that word with a grain of salt) it's almost more like a lottery. Once the omega has been pregnant for three months, a prenatal paternity test is used to determine which alpha's sperm took, and thus the omega is given to that alpha as their mate.
Techno hid his omega status for a very long time but is eventually found out and forced into the program, much to his dismay. Quackity is a nepo baby slash mob brat alpha who is just elated to hear his seed managed to knock up such a pretty omega <3