Can you explain Sai Baba's teaching on oneness with all creatures, particularly the story involving the dog and the pig?

In Chapter 9, Sai Baba imparts a profound teaching on universal oneness. When a lady devotee expressed that she was dependent and couldn't serve him directly, Baba explained that he had been fully satisfied when she fed a hungry dog. He revealed, "that dog which you saw suddenly, I have oneness with him." He extended this to a pig as well, stating his oneness with it. This astonished the lady. Baba then clarified his divine nature, saying, "Sometimes I am a dog, sometimes a pig; sometimes I am a cow, sometimes a cat... in such forms I roam." He instructed her to renounce the sense of difference and see him in all beings, as that is the truest form of worship. This teaching, described as "supreme nectar," moved the lady to tears of joy.