WARNING: The following contains major spoilers for the Demon Slayer: Kimetsu no Yaiba manga, by Koyoharu Gotouge, available in English through Viz Media.
Despite being the Progenitor of Demons in Demon Slayer: Kimetsu no Yaiba, Muzan Kibutsuji generally feels little more than disdain toward his creations. He has a few favorites among the Kizuki such as Rui (Lower Five), Gyutaro (Upper Six) and Akaza (Upper Three), but perhaps surprisingly, an official Demon Slayer "fanbook" written by the manga's creator, Koyoharu Gotouge, revealed that Upper-Rank Two, Doma, is a demon whom Muzan doesn't really like. As Lower-Ranks Six, Four, Three and Two can attest, Muzan is not averse to eliminating demons he's unhappy with, even if they're Kizuki. So how did a demon who is not favored by Muzan become Upper-Rank Two -- and why doesn't Kibutsuji just kill him anyway?
Doma, aka the bane of Akaza's existence, makes enemies everywhere he goes despite being by far the friendliest and most polite of all demons. In Akaza's case, his hatred for Upper Two stems from the fact that Akaza is a founding member of the Kizuki, while Doma joined much later but quickly surpassed Akaza in both power and ranking. Kokushibo, Upper-Rank One, treats Doma as a subordinate who needs to be lectured, and Nakime curtly declined Doma's offer for a romantic rendezvous. It really could be as simple as the fact that Muzan dislikes Doma because he finds him just as annoying as everyone else.
When Muzan called the Upper-Ranks together in the wake of Gyutaro's death, he angrily questioned why none of them had eliminated the Demon Slayer Corps or located the Blue Spider Lily. While Doma offered to gouge his own eye out in repentance for having recruited Gyutaro, he also admitted that he never found the Blue Spider Lily or Ubuyashiki because he's no good at detective work. Surprisingly incompetent for the third most powerful demon in the world behind only Muzan and Kokushibo, Doma couldn't even get through the first gathering of the Upper-Ranks in 113 years without annoying Akaza into punching off parts of Doma's head -- twice.
Muzan Kibutsuji certainly seems like the kind of man who would kill someone for the high crime of being really annoying. Yet strangely, Doma is left free to do as he pleases -- running his house of worship called the Eternal Paradise Faith wherein he eats his followers so they may live forever as part of him. Doma's penchant for eating women may play a big part in why he became so powerful so quickly, as he once commented that women are extra nutritious for demons thanks to their ability to nurture babies with their bodies. Doma's power as Upper-Rank Two of the Twelve Kizuki is most likely what keeps him safe from Muzan's murderous wandering eye.
At the end of the manga's "Swordsmith Village" arc, Muzan revealed that he never wanted to create other demons in the first place. The only reason he did was in hopes that one demon would become powerful enough to withstand sunlight. If Muzan were to consume such a demon, he would gain the power for himself and achieve his dream of becoming the perfect being. The most likely reason that Muzan never killed Doma is that he couldn't risk losing a demon so powerful. Muzan likely believed it possible that one day, Doma could become strong enough to survive sunlight, so killing him might've been detrimental to the reason Muzan made him a demon in the first place.
When Muzan slaughtered most of the Lower-Ranks, he likely did so confident that none of them could ever be powerful enough to withstand sunlight. Doma, on the other hand, had too much potential to justify killing him just to get him to stop talking. Once Nezuko Kamado became immune to sunlight, Doma was surely kept around for the inevitable conflict with the Demon Slayer Corps, in which Doma succeeded where even Akaza failed by killing one of the Hashira, Shinobu Kocho. At the end of the day, Doma's potential outweighed his annoying personality, even if only barely.
