Are all the Avatars essentially the same person?

Question

Are all the Avatars the same person? We know that all the Avatars are different incarnations of Wan. When the Avatar dies, but when they are reborn, the next incarnation experience different things. But how is that the Avatar can remember things from their previous life, For example... When Korra encounters Iroh in the spirit world in 'A new spiritual age', she instantly recognises him, stating ''I know you...Iroh''. But why does she recognises him. Is it because her soul recognised him from when she was Aang.

There many other examples that might support that they are the same person

• 'We will be together for all your lifetimes' -Raava to Wan

• 'My teacup' - Korra to Iroh ''A new spiritual age''**

• 'You think I am weak!? I have mastered the elemenents a thousand times in a thousand lifetimes' -Roku to Jeong Jeong ''The deserter''

• 'She is connecting with her deepest Avatar memories. She must confront her own past...'" -The Shaman on Korra, "Beginnings Part 2" '

• 'It was yours, long, long ago. You used it to carry the light spirit around before the two of you became one.' -Iroh to Korra

• 'My past. Our shared past' -Roku to Aang, "The Avatar and the fire lord"

Answer 1

No. All the Avatars are not the same person. Each is distinct and unique. However, they are all carrying or being carried by the vastly powerful elemental spirit of Raava, the entity which makes the Avatar capable of bending multiple elements.

• Think of the past Avatars as a form of spiritual memory carried by their relationship to Raava. She retained their spiritual memories as each Avatar died and this became a potential cache of memories that a new Avatar could access through meditation, periods of enlightenment or through the Avatar state.

Many years later, as an elderly Wan lay dying on a war-torn battlefield, Raava reassured him that all would not end in vain as they would remain together for all of Wan's lifetimes and would never give up. As Wan exhaled his last breath, her spirit, accompanied by Wan's, exited his body and reincarnated as the next Avatar. --Hedrick, Tim (writer) & Graham, Ian (director). (October 18, 2013). "Beginnings, Part 2". The Legend of Korra. Season 1. Episode 20. Nickelodeon.

• Since Korra's battle with Vaatu, she has lost her connection to the spirits of the previous Avatars and potentially any memories they may have had. Korra's first meeting with Iroh, was before she had lost her connection to previous the previous Avatars and while she was in the spirit realm which gave her a potential connection to information only possessed by her previous incarnations.

• Korra (and Jinora) enter the spirit world. Here she meets Iroh for the first time. Her feelings tell her he is familiar. Given his distinctive appearance and legendary status, she may have simply put the two and two together. Remember, she would have been briefed on the previous Avatar, Aang, given his accomplishments and would know about the stories from her time with Tenzin. He then confirms her guess, and identifies himself as a friend of the former Avatar, Aang.

Answer 2

Yes, all incarnations of the Avatar are in fact the same person over multiple lifetimes. This is backed by the wording of the show itself, referring to the process as 'Reincarnation' rather than 'Possession'. A lot of your own sources in the question itself work to back this up. I will also be referring to what -for the time being- I would like to call the Name, Body, and Spirit that has come to be known as 'The Avatar'.
"We will be together for all your lifetimes." -Raava to Wan.
This first sentence requires us to understand two things: Wan has had lifetimes before this one, where he had a different Name and Body but still the same Spirit, this Spirit being a human one. And every lifetime afterwards Raava and Spirit have combined to form 'The Avatar Spirit'.
Wan is known as the first Avatar and every subsequent lifetime he has is also the same 'Avatar Spirit'.
"My teacup." -Korra to Iroh.
The word here is 'My'. Not Wan's teacup. Not the Avatar's teacup. But My teacup. Even though they may have a different Name and Body both Aang and Korra (along with any lifetime succeeding Wan) would refer to this as their teacup.
"It was yours, long, long ago. You used to carry the light spirit around before the two of you became one." -Iroh to Korra.
Again, the phrasing used is 'you' and 'your' as apposed to Wan's. This is because Wan and Korra are only Names with different Bodies in order to describe the same Spirit. I would also like to take this chance to reiterate that Raava and Wan (two) came together to form The Avatar (one). In this case spirit(Raava) and Spirit(once called Wan) became 'The Avatar Spirit'.
"My past. Our shared past." -Roku to Aang.
These two sentences have all of the information about reincarnation within them. The fact that Roku is saying this to Aang actually means nothing, as both are just different Names and Bodies. These same words could be said between Aang and Korra, as well as Kioshi to Roku, Wan to Korra, and to push the point home even the lifetime that came directly before Wan all of the way to Korra. The fact is that in this moment Roku is saying this to himself. A being with a different Name and Body but the same Spirit. We can even go ahead and change the quote itself to "Our Past. My shared past." There are not 10,000 Avatars but One Human living 10,000 different lifetimes. Lifetimes where the Name changes, the environment changes, the nationality changes, gender is a part of a Body that changes. Only the Spirit that encorporates all these changes remains the same, learning from each subsequent lifetime but never becoming anything more than one(Wan).

What you should take from this is that 'The Avatar' in the show is actually a representation of you. You are a collection of all your past lives come together to form the person you are today. Your name and body have been given to you, but your spirit is the 'You' that existed to receive them.
Even in the show we see that when an Avatar dies ("Beginnings Part 2") that human's Avatar Spirit leaves behind the Name and Body to continue on to rebirth as per the Reincarnation Cycle. This is to be understood that Every human Spirit in the show goes through this same rebirth cycle, the Avatar being 'special' due to Spirit mixing with Raava (spirit+Spirit=Avatar).
'You' are special too, because 'You' are the same as 'I' and I Am special. Where I said Name, Body, and Spirit before I now say Ego, Id, and Superego, the things we all have that come together to form each individual's 'I Am'. These kinds of thoughts are very Metaphysical and 'Deep' as they ask impossible questions such as 'Life after Death?' And I hope that the parallels between this series and Real Life have enlightened you as they have me.

Afterword:Hello, this is my first time answering and I could not get the coding to work as I wanted. My intention was to put Blockquotes around the show quotes but could not find how to 'stop them'. Any edit or comment on how would be appreciated. Also my last paragraph may have drifted into 'Opinion' but is still deeply connected with the theme of the show and question both, so I have deemed it necessary. Thank you.