This “trope” which is noticeable in Japanese media is a result of the nuances of Japanese style swordsmanship, especially from the Edo period onwards.

Unlike sword and scabbard system in many other cultures, the swords of Japan have two components that have led to this type of action that you see consistently in modern media: 1. The interface of the blade collar with the scabbard throat. 2. The art of iaijutsu or drawing of the blade.

Looking first at the construction of Japanese swords especially from the Edo onwards, you have the blade surrounded at the base with a collar called a Habaki, usually made of copper which enables the sword to not just sit in the wooden scabbard (saya) but lock firmly in place and prevent unwanted intrusion from things like moisture.

On the scabbard side, the mouth of the saya is often covered with a protective ring called a Koiguchi. This can be made from different materials depending on the wealth or status of the owner, but it is commonly made from Buffalo horn. The saya with its koiguchi will be custom fit to securely hold the tanto, wakizashi, or longsword (katana). Most other cultures either do not have a blade collar, use friction fit on the blade in the scabbard, or use metal throats on their scabbards.

During the Edo period this relative time of peace allowed many samurai to develop schools of martial arts, practice fencing, and engage in duels. One of the more popular disciplines to come out of this time was iaijutsu or the art of drawing the sword, in an attacking action, which continues to exist today in the practice of iaido. During iaido, in order to “unlock” the blade from the aforementioned securely tightened system, the first step is called nukitsuke. Part of this is while holding the saya, using the thumb to apply pressure on the guard (tsuba) to gently release the sword from the saya. This will prepare the blade for the more violent action that is to come.

Depending on the size of the blade, the tightness of the fit, the haste of the situation, and the competency of the practitioner this release from the thumb could be seen as a single discrete action or incorporated more seamlessly into the rest of the draw.

This is sometimes taken to exaggerated lengths in media; you point out the concept of “cocking a gun” used for implied violence. In the same way, a samurai who unlocks his sword would be informing anyone who saw him (invisible audience or otherwise) to be prepared for his blade to be unleashed. There is a bit of a historical parallel to this, again from the Edo period, the concept of koiguchi san sun, or three “sun” (9cm) from the koiguchi. If a blade were pulled bearing more than 9cm it was considered officially “drawn” even if it was still technically in the scabbard, and if this were done in the wrong place or time the samurai was subject to forfeit titles, estate, have all of their family made ronin for generations to come, or even sentenced to death.

You also note that the re-sheathing of the blade is done slowly and deliberately, which is another borrowed element from of iaido. Just as the sword is initially drawn deliberately so as to not cut through the saya or damage the koiguchi, so too is the blade returned slowly without slamming it against the horn collar, applying only the required pressure to secure it in the lock.

Another concept which is considered a samurai discipline which occurs during the noto or sheathing of the blade is called zanshin. This refers to a posture of readiness for both mind and body and to not be negligent to attack even if you have felled your immediate adversary, preparing for any additional threats.

This could be seen as drawing and attacking quickly, but slowly putting away your blade while still being prepared for additional attackers. In media this perhaps will be exaggerated, so show a disregard for the enemies slain while ceremonially putting away the sword.

References:

The doctrine of Iaido by Matsuoka Yoshitaka, 8-dan kyoshi