Like many others, while not exclusive to Japan, this trope is very commonly used in Japanese Media.
In Japanese culture, omoiyari ("empathy", "assuming good intentions", "humility") is a very important virtue which asks that every person think of the position of others before they speak or act. This ideal is applied in a variety of ways, but in a combat scenario, it's almost always portrayed as someone who is many levels above their opponent using only the bare minimum of force or power to defeat the weaker fighter. This is why many characters (even when the stakes of losing are high) do not open battles with their best moves or strongest form; technically, this trope is a token effort to save face for the opponent, so that they don't look incompetent.
Of course, it's usually an Oh, Crap! moment if the character is holding back and the weaker character still gets curbstomped. As such, the gesture doesn't actually have to mean that the character cares about the reputation of the weaker opponent. Some Faux Affably Evil villains do this precisely because it's more humiliating to lose in this manner, and the villain gets to superficially look civilized or elegant in the process.