I reviewed episode 7, Showdown! Inuyasha vs. Sesshomaru, which is the episode where Sesshomaru loses his arm. During the fight, Inuyasha doesn't just cut off the arm; he slashes up the whole length of the arm (Sesshomaru was in dog form at the time), likely damaging it extensively. In fact, I didn't see the arm afterwards. I didn't see it fall off and I didn't see it on the ground at any point afterwards, which might imply that it was disintegrated. Even if it wasn't disintegrated, it's likely very badly damaged.
Now, on to whether it's even possible Sesshomaru's arm could come to life. The answer is maybe, and it goes back to the Great Dog Demon, his father. The Tesseiga, Inuyasha's sword, is actually made from the Great Dog Demon's fang, and the Tenseiga, Sesshomaru's sword, is derived from the Tesseiga. So, just like Kirinmaru, the Great Dog Demon did lose a part (a fang). Whatever happened in this case is going to be more applicable to Sesshomaru, given we're talking about his father here. And what is the case with the Tessaiga and the Tensaiga? Actually, it seems both do possess some sentience.
From the Tessaiga's wiki:
The Tessaiga also seemed to possess a mind of its own, as it has been seen guiding Inuyasha, such as wanting to be drawn or giving him advice in certain situations; this was a trait Tessaiga shared with Sesshōmaru's Tenseiga.
From Tenseiga's wiki:
Tenseiga had a slight form of sentience, as there could be seen at various points where it was communicating with Sesshōmaru, and expressed to him things such as a desire to be drawn or prompted him toward certain actions for the benefit of others; namely young Rin.
Much like the Tessaiga, Tenseiga shows some degree of sentience. As it sometimes requested to be drawn, or in 3000 Leagues in Search of Father, Tenseiga told Sesshōmaru to save the life of Kanta's father.
I believe the two swords primarily communicate by pulsing, which is then interpreted by their wielders. I'll admit I don't feel particularly inclined to scour all of this very long show to find these moments. Let's just conclude here that these swords came from the Great Dog Demon's fang, and they might be alive in some capacity.
Now, I've heard it said that the reason Sesshomaru couldn't reattach his arm or regrow it was due to it being cut by Tessaiga. The author even said something to this effect, though I can't find the interview to check what exactly was said:
From Tenseiga's wiki:
According to Rumiko Takahashi, Tenseiga could not heal Sesshōmaru's severed left arm because it was cut off with Tessaiga.
The citations say this come from page 172 of Zusetsutaizan Ōgikaiden, which seems to be one of those fan-book things and was called InuYasha Profiles by Viz when they released int in English.
What does this mean as far as Sesshomaru's arm taking on a life of its own? Is it not possible because it was cut off by Tessaiga? Actually, if I'm not mistaken, we have an answer for that! That's because Kirinmaru's parts were also cut off by Tessaiga! Tessaiga was an originally a weapon used by the Great Dog Demon, and he used it (or a sword that looks like it to me, at least) to cut off Kirinmaru's arm and horn during their battle. Unless I'm wrong and that wasn't actually Tessaiga (which is a real possibility, as I'm not extensively familiar with Inuyasha lore and when things happened), this means at the very least that being cut off by Tessaiga is not a reason that Sesshomaru's arm couldn't come to life.
The body parts of dog demons might come to life when they're cut off, or at the very least they might have some sentience, but we can't really know for sure. Sesshomaru's arm might have been too wrecked from that fight to take on a life of its own afterwards, even supposing that's a thing it could do in the first place. Frankly, I don't believe the world of Inuyasha is logically consistent, so the whole discussion might be moot.