No, loan sharks are not legal in Korea, East Asia, or anywhere for that matter. And this leads us to your second question, because it is not legal, unlike most debts that die when the principal signatory (in this case, Wonwoo's dad) dies, illegal debts allow lenders to hunt down next of kin to ask them for the payment instead (in this case, Wonwoo and his siblings). Though Wonwoo initially avoided legal debts by renouncing his inheritance, his dad's illegal debts will continue to haunt him as that is what illegal loaning has to do in order to enforce payment. Basically, since its illegal, there is no legal court order and punishment to force Wonwoo to pay the loansharks back, instead the loansharks resort to violence, coercion, and threats to make their debtors pay. But when discovered by law enforcement, loan sharks will face a fine of $34-40,000 and at least a 1-12 months, or more if the case is severe, imprisonment. At least, that's what Korea's law says. Hope this helped. :3
You can read more abt it here:
https://www.npr.org/2021/10/20/1047735128/the-underground-world-of-debt-collection-in-south-korea
And here:
https://share.google/Xgd6nHgNyPuL6nsxu

So, is loan sharking actually legal in Korea? And more importantly, is it really normal for children to end up paying their parents’ debts? Could someone please explain how this works? Thanks