The Self is the eternal, never-changing One.
Anything that changes, such as the body and mind, can never be the Self.
Once we realize our true Self, we will see the same Self everywhere and experience the Oneness.
There is a story from the Upanishads that tells about the king of the gods, Indra, and the king of the demons. In the story, Indra represents the refined, more spiritually evolved individual, while the king of the demons represents someone who has not yet purified himself. It goes like this:
Once the king of the gods, Indra, and the king of the demons went to the same teacher. They both asked the same question: “Master, please tell me who I am.” The teacher gave the same answer to both. He said, “Go to the well, look down, and see what is reflected in the water.” So they both went to the well, looked, and each saw his own physical reflection. Immediately, they both said, “That is me!” and went home. The king of the demons started decorating, adoring, and indulging his body in every way. He ate well, slept well, and allowed all the senses to enjoy whatever they wanted.
Indra, however, began to question: “How could I be that? Yes, I saw the reflection of my body, but I have read in the scriptures that the ‘I’ never changes. It is eternal, ever clean, ever pure. It is never born, never dies. But a body is born, it dies, and in between it changes every minute. How could that be the Self? I think I made a mistake. I did not understand him correctly. Let me go back and ask him.” By this time, several years had gone by. Indra went to the Master again. The Master said, “Who is asking this question now?” “Well, the mind,” Indra replied. “Then, that is you. Now go!” Indra went home. Again he started thinking and thinking and thinking. Twelve years or so went by. He analyzed: “I don’t see how I can be the mind either. It often gets very crazy. It’s not always the same. It fluctuates, gets excited, gets depressed. It forgets. There are lots of changes in the mind. That cannot be the Self. I think I should go back to the teacher again.” This time, the Master said, “Who is watching the mind going through all these changes? That is the Self.” Indra went back, contemplated, and realized, “Yes, that never changes. That is always the witness of everything. That is who I am.”