Both are fantasy romances between a god ML and a human FL and they have somewhat similar storyline.In both cases, ML is being punished by a deity and forced to wander the world eternally until FL comes with the power to kill him.
In both, we get to see glimpse of their past lives, set in medieval Korea.
Both, feature a mix of bickering/humorous scenes and more angst filled moment ( they will fight over the ML being an inconsiderate jerk before crying and swearing eternal love while being torn appart by fate).

A tale of power, ambition and corruption

The plots may seem very different, however they have some of the same noir, thrilling elements to it
Both are dramas that feature chaebols and normies, plagued by ambitions, who are playing dirty tricks on each other to achieve their goals. In both cases, the ML is an outsider from the corrupt chaebols ( in Doctor Prisoner, ML is a doctor that wants to destroy the people in power and get revenge, in Impossible Heir, ML wants to steal the company from said people in power )
Both have tense suspenseful moments, business tricks that shock the opponent, race against the clock to one-up the villain.

Nevertheless, there are also a lot of differences : there is no cute romance in doctor prisoner, don't expect kisses and confession. It is also darker in my opinion Than the impossible heir

When romance is thrown in the turmoils of history.
Both dramas start like your typical romcom, boy meets girl, girl dislikes boy... (Granted, in My Dearest the FL is wearing a hanbok while in Youth of May, she's running around in jeans.)

But then, history with an H intervenes. For Youth of May, the dictature and the shadow of the Gwangju massacre are looming over the happiness of our protagonists. For My Dearest, it's the Qing invasion that sweeps them away. So, if you like a *somewhat* realistic period piece that isn't all political and that still has time for some cutesie lovey dovey stares between the lead, these dramas are for you.

In both you have the perfect balance of lightheartedness and Melo

Both shows revolve around not so normal high school students.
Both shows feature some sweet slight romance between traumatized kids ( oh the angst ! )
Both shows are fast paced action thriller.
Both FL are driven by revenge while the ML kinda just wants to help.

Both are dark, violent and grim tales, one of revenge, another of corruption. If you loved to watch Han So Hee hack her way through a small army of gangsters in my name, you'll similarly enjoy Namkoong Min shoot every spook in Seoul.

Both dramas are visually stunning and have hypnotizing osts and overall similar bloody-vibes.
Both are action-packed thrillers

Both are action-packed shows set during the Japanese occupation of Korea. Both have an enjoyable romance subplot in the mist of all the fighting and killing.
Both feature badass fighter female leads.

While 'Gyeongsag creature' is more horror themed, 'Song of bandits' has a strong western vibes.

Both are gripping revenge stories. Both are thrilling with the same somber feel. You get the grim satisfaction of seeing the bad guys lose their shit and despair when they are ultimately punished.
The visuals and soundtrack in "the Glory" are a bit better but "reborn rich" is still stunning