Since the beginning, he's is the Reincarnation of Indra, so that means Indra was already within him from birth. He wasn't possessed by him like some kinda ghost.
If he was Indra's reincarnation from birth? Why his deeds didn't match Indra's when he was a child. If he was Indra himself, Why he changed a little, just at beginning of his life and got back to his own theories and views once he matured? And that too all of a sudden?
Therefore, Madara's soul = Sasuke's soul?
Indra and Ashura are reborn every time a great rivalry is born from two powerful people who will greatly affect the world they live in. Madara was never born with Indra's soul/spirit in him, he only shared Indra's eyes, Hashirama only shared Ashura's ability to find his strength in others, and value in people's lives.
Once Hashirama declared Madara his rival, when they were younger while competing in a stone tossing game thats is when Indra's and Ashura's rivalry and souls were reborn into them. If you know what happened afterwards, they ended up changing history, though they never got peace.
The rivalry stopped once Hashirama died, and the soul of Indra and Ashura were reborn again into Naruto and Sasuke when Naruto declared Sasuke when they were kids after sasuke's parents were killed. Just like Hashirama, Naruto felt a great bond,rivalry and hatred towards sasuke. Sasuke is Sasuke, Naruto is Naruto but the rivalry the two share are that of Indra and Ashura.