They have the same feel, psychological manipulation. Both are villain origin story and features psychopath.

The cinematigraphy and the use of music is similar in these two. Top notch things. Really different plot, though.

Both are about survival game with one mastermind.
3Nen has a really great theme it focuses on, it is about bullying. It is a really good japanese-style-dark comedy and full of life lesson.
365 is a really suspenseful, similar to manga Btoom.

Both tells a tale about everyday life and the leads have depth and doing reflection all the time. Both are healing dramas.

They are the same in the sense of vibe, and both are a story of professional worker. While Black Dog has this warm, healing feeling, Diary of Prosecutor is more quirky and portray thestruggles of adult worker life in Korea.
They air at the same period too, so I watched them hand in hand and I really liked both.