Two recent k-dramas have brought up the subject of baby kidnapping and in both cases, it was the family of the father that was prepared to steal the baby away from a single mom and discredit her in an attempt to gain custody.
In "Heard It Through The Grapevine," Lee Joon's character Han In Sang has a baby with Go Ah Sung's character Seo Bum. When he finds out that she's expecting, he wants to marry her but his ambitious parents don't see her as a good match for their son and start scheming to take her baby away. They don't think her social standing places her in the same realm as their peers and they want to tightly control any situation that might affect their political influence.
So far they have just worked hard to separate the couple and keep the new mom from her baby but it's only the first step in their plan.
The same thing happened in "The Greatest Marriage" when Park Si Yeon's baby daddy, played by No Min Woo, let his family take their child away. They accused her of stealing from their son by having his child. Since she already had custody they worked hard to ruin her career and reputation so they could claim she was not a suitable parent. They also physically kidnap the child.
And another attempt at "baby-napping" might also happen in the other new drama "Fool's Love." Uee's character Do Do Hee has a baby with a young successful lawyer, but his parents are also ambitious and pretentious. They do not want their son's reputation ruined by an illegitimate pregnancy. They may scheme to do the same and Choi Woo Sik's character may get caught in the middle.
Baby kidnapping or attempts to gain custody at a mom's expense are not specific to Korea. People do mean and crazy thing to obtain children's custody everywhere. However, k-dramas have used this plot device a lot lately. There's little meaner than stealing someone's baby but there may be a cultural precedent for it happening in k-dramas.
Since pregnancy outside of marriage is still frowned upon in Korea, some see it as an advantage for the father's family to raise the child. It can help avoid the social stigma, offering benefits for the child's future. A k-drama wife may occasionally claim the child of her husband's mistress to diminish the potential scandal. That was the case in "The Heirs" when Lee Min Ho had to call his father's wife mother.
In any society that was traditionally patriarchal, children belonged to the fathers and dads received custody in the event of a break-up. That has obviously changed in Korean daily life but the changing perspective continues to interest k-drama audiences.