Big Mom's DF has a nen-like condition/restriction that prevents her from taking a soul if the person isn't afraid of dying. Since Kaidou isn't afraid of dying, BM's power doesn't work on him. Moria, on the other hand, can bypass immortality if he takes your shadow. This is why Kaidou chose Moria to fight him, but Moria refused to use his DF. Kaidou then hit Moria with an uppercut so powerful that Moria's chin fell off and his neck was lengthened.
Kaidou holds on to Moria's chin in hopes that one day he can extract the kage kage no mi from it like how Vegapunk replicated Kizaru's fruit.