The spirit of the moon was split between Yu and the fish when it used it’s powers to save her. So the moon spirit resided in two beings at once. The Avatar spirit could reside in two beings at once. Remember when Aang spirit bent Ozai, we saw their two spirits competing for control and we knew that he risked losing control to Ozai. What if when Aang used spirit bending on Yakone while in the Avatar State, he ended up losing a chunk of the Avatar Spirit inside Yakone, which gave Yakone spirit bending powers. So the Lion Turtle transferred the power to Aang by spirit bending him but allowing Aang to ‘win’ and letting the power flow into him. Then when Aang used the power on Ozai he prevented him from ‘winning’. What if Yakone ‘won’ a chunk of the Avatar Spirit, and the ability to spirit bend, and then became Amon.
EDIT: Also, it would explain why Korra cannot go into the Avatar State. She should’ve gone into the Avatar State several times now, but especially while being blood-bent. When Amon was being blood bent, we saw his legs (but not his eyes) and he faltered for a moment then continued, similar to how Aang responded to blood bending.
It’s possible that the reason Amon wants to make a big spectacle out of his showdown with Korra is that he wants everyone to see his ability to go into the Avatar State and Korra’s inability to do that, and declare himself the true Avatar, which would help people follow him ideologically. It’s also possible that having a broken chunk of the Avatar Spirit is why Korra is so bad at spirituality.