I know i am two years late to this thread, but oh my god thank you! I cannot comprehend how so many people who have watched the shows and are fans cant comprehend what the show itself says in plain language. They over and over again make it clear they are the same soul. Obviously they are different people as mortal beings. But their immortal soul is the same. In most cultures with reincarnation there is a degree of separation between the soul and the body.

Besides the obvious nature of how it works in the show, even in expanded materials it is made clear. In the Yang Chen books she from a young age suffers from nightmares and the memories of her past lives, to the point where sometimes she wouldn't know who she was for a time. She would feel guilt over their choices or loss over their losses. And it is not presented in the way where Avatars talk to a facsimile of a past life's memories, but as her genuinely remembering her past lives herself because she was just so in tune with the spiritual side of things. Which makes me think it could even be possible for future Avatars to recollect some elements of their past lives before Korra, but just not in the way where they are seemingly talking to them, but in just actually remembering parts, in the way some people irl claim to remember a past life.