Drops of God

The story revolves around Kanzuki Shizuku, a young man who was estranged from his father, a world-famous wine critic. When his father passed away, he left behind a wine collection worth a fortune, but in order to inherit it, Shizuku must solve a series of puzzles left by his father to identify 6 wines. His competitor is Tomine Issei, a prominent wine critic, whom his father adopted before his death

Both dramas revolve around a girl who was as gifted as her father. As a child they got separated and they did not have any contact. Both girls try to avoid being confronted with their inherited gift. Both women's fathers are very successful and well known for what they do. Then a death due to a serious illness brings them back into the orbit of their father and both they have to find a way to proof their gift.
In both dramas these girls are meeting old acquaintance of their father who teach them both all that they know.
There are additional similarities in regards to family dynamics but these would reveal too much.

Still it is basically the same story just in Drops of God the story revolves around wine, while in Miss King it revolves around shogy.

Both of these dramas feature a sommelier/wine critic male lead, with a female lead who must test her own abilities against his. The two leads have a mutual connection through a a third party, and both dramas feature an undercurrent of mystery.