TV series made in South Korea
Does the phrase make sense?
Yes, it makes perfect sense. As you stated, though, it is a phrase, not a complete sentence.
Are “South Korea’s TV series” and “TV series of South Korea” synonymous with “TV series made in South Korea”?
Yes, these phrases can be synonymous. I prefer the third one, though, as it is the clearest. TV series of South Korea could also mean TV series about South Korea – not necessarily made there. Thus, it could be confusing. Another option is: South Korean TV series.