If you wanna turn this into powerlevel stuff:
Kaidou said he see Oden's visage/appearance in the scabbards, Kaidou said Oden's presence in the sword while BM noticed the sword instead of Zoro's haki. Surely BM-Kaidou will recognize / mention Zoro if Zoro is the source of energy imbued in Enma (source of threat/danger). And seeing Oden's face in his followers that participated in legendary hour is imo just Kaidou being reminiscent and anxious PTSD whatever, causing his defense to drop etc. Since Kaidou had said "they didn't have Oden's haki so why can they pierce me?" but Kaidou mentioned that "why is there Oden's presence in that sword that can pierce me?" instead of the person doing it unlike when he mentioned the scabbards' person.
Kaidou sense SNs "voices" after Hakai, is it literally their voice? No it's their presence / lifeforce / energy / haki. So to mention someone else's presence in the only sword that became the threat to him instead of Zoro's other sword that is also can be imbued with CoA but having minimum effect, it's a direct opposition from mentioning Zoro's own CoA as the threat.
Since Zoro can always imbue his other sword with CoA and it should cause exact same damage or threat, but Kaidou doesn't acknowledge any of Zoro's attack that used his other swords (still imbued by CoA) to have the same level of danger. Only Enma.
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My conclusion, Enma store Zoro's haki and exudes it as its own. But the same thing also happened to Oden as he was also the wielder of Enma for quite some time. Thats why Kaidou recognize Oden's presence more than Zoro's, and thats why BM acknowledge Enma in Hiryu Kaen while not acknowledging Zoro's CoA in any other of his attack.
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Back to the topic: and thats why Zoro's powerup in completely mastering Enma against King will give narrative significance of inherited will.
