It is partly true. They are the same person and at the same time they are not. I agree and I'm pretty sure that the original heart and the new heart of the nobody merge and become one when the person is restored as a human (or somebody), it is the only logical explanation for Axel and normal nobodies. Roxas had his own heart compartimentalised inside Sora's during KH3 instead of being fully merged but he is a very special case as he drawed from both Sora and Ventus when he was born as a nobody but without inheriting any of their identities or full sets of memories so he remained as a separate entity. Similar with Xion (save for the particular details of how she was created) and most likely Naminé, although Naminé was never given much exposition and we don't know for sure how many memories she received from Kairi or Sora, if any. We don't even know if there was another heart inside Kairi (Ava or someone else?) that determined how Naminé looks like but we'll know in the future if that's the case.
About the differences between the original and the new heart I can only speculate and I don't know to what extent this would apply to lesser or special nobodies such as Roxas, Naminé or replicas); but I'm convinced that the new heart of those nobodies who retained human form and 'showed signs of a burgeoning replacement' doesn't differ much in nature than the original heart. Why? Because they remember who and how they were as humans (Axel, Saïx, Vexen, Zexion, Xaldin, Lexaeus) and even if they don't fully remember, at least they remember what having emotions was like (nobodies of Union X people: Larxene, Marluxia, Demyx, Luxord).
I'm not completely sure how or if being norted affects all this, but it is kinda implied that the piece of Xehanort's heart in the norted nobodies prevents them from developing a new heart or at least slows it down. Xemnas stated in DDD that Xehanort didn't want the organization members to became too aware of themselves (members who had not been norted yet) obviously because that would hinder his plans.
Anyway I'm not entirely sure I fully understand how being norted works, because some characters seem to be more affected or possessed by Xehanort's heart despite having their own heart to oppose Xehanort's (Terra) than others who retain their wills despite being norted immediately after becoming nobodies again and had no time to grow new hearts and thus they had only a piece of Xehanort's (Saïx and KH3 norts who were not time traveling hearts in replicas) and it doesn't seem that sheer willpower is enough to effectively refrain Xehanort's heart from completely taking over because if that was the case Terra wouldn't have succumbed to that extent. I've come to the conclusion that when someone is norted on their own will, the piece of Xehanort's heart doesn't have to take over completely and the subject is allowed to retain his free will (because no way norted KH3 Vexen or Demyx are more strong willed than Terra even if Terra's own darkness made him an easy target) Although Terra received a full dose of Xehanort's heart and not a piece so there's that...
I won't say anything about Vanitas because he is not a nobody but a norted dark half of Ventus having his own heart but willingly serving Xehanort, and Xigbar was playing all along and acted on his own without being affected at all despite being a norted nobody (didn't have Luxu's heart... unless he never became a nobody for real) I guess either Luxu somehow retained his heart, he had it restored later or he didn't need it to still consider himself Luxu and knew how to keep Xehanort's heart at bay. Who knows. But I digress and norts are not the topic, I'll stop that here.
The thing is, if nobodies draw from their past experiences in order to know how to act when lacking a heart, they are still attached to being themselves somehow, even if the absence of the heart makes them prone to easily act with cruelty or disdain as they are unable to feel remorse, at least until the new heart begins to manifest. And the new heart likely develops faster when the nobody tries to bond with others instead of acting as a selfish sociopath, that's why Axel was the most obvious one to have grown a heart through his friendship with Roxas and Xion. The new heart is born as the nobody interacts with others and the world, but doesn't prevent the nobody from remembering their past or how they felt in the past, meaning their identity has some continuity.
The fact that nobodies act differently than their human selves can be interpreted as them being different altogether, but I'm more inclined to view that discrepance as an adaptation to their new and confusing circumstances as nobodies and not as a radical divergence in nature, specially after knowing that the members of the original Organization were deceived and tricked into thinking that they had no hearts and that they had to create and artificial Kingdom Hearts made of people's hearts in order to be recompleted, and they were assigned missions to achieve their (ultimately fake but still dangerous) goal as part as one of the multiple Xehanort's experiments on hearts when they actually were being kept busy until the day Master Xehanort norted them. They were being fooled by the same man who gave the orders and then some of them deviated and started having their own agenda but in the fist place that derived from their allegiance with the Organization. They were recruited short after becoming nobodies and from that moment they didn't fully act on their own without Xehanort influencing them one way or another, it's difficult to say how they would have been as nobodies without someone mannipulating them.
Whether the past self, the nobody or the recompleted person can be considered the same person or not is debatable and there is not a clear answer as it's actually a deep philosophical question and I'm no expert, but from an external point of view it can be seen either way. But when it comes to any kind of identity one must always respect what that person feels, that's the most important thing to have in mind (I know, Axel/Lea is a fictional character, but you know what I mean). For me the nobody and the human self are essentially the same being as long as there is an identity continuity: the nobody sticks to their past self memories as human if he/her remembers it, and the recompleted person sticks to their full past as human and nobody. That's the case with recompleted Axel, he is fully aware of his identity, accepts his past as a human and his past as nobody and both are part of himself. He doesn't have a split personality or something like that, so we must consider him as one and the same.
The Union X characters that ended up in the Organization are implied to not fully remember their past so we don't know if they have an identity continuity at all. Nomura said Lauriam is Lauriam and Marluxia is Marluxia, but I think that is because Marluxia doesn't identify himself as Lauriam, either because he doesn't remember at all or because he turned his back on his past until Sora defeated him in KH3 and made him remember who he truly is, moments before Marluxia vanished. That does not apply to Axel.
