Alright, so, I think the problem here is that a lot of people tend to view the polarities in Axel's personality as if they ought to be mutually exclusive, whereas the games make a pretty strong case for why Axel is the way he is. As
@Face My Fears pointed out, in CoM we view Axel through the lens of characters who have cause to mistrust him, and against whom Axel has cause to perform the part of an antagonist-- but we never learn
why he behaves the way he does, and the game leaves us with more questions than answers as to what end his actions were intended to play out. What we learn about Axel in CoM are certain qualities of character: he's ruthless in pursuit of his agenda, he's highly intelligent and coldly at ease with manipulating others, but he's flexible in the degree of his antagonism depending on the situation: he makes it clear that he and Sora aren't actually enemies, which signals that there's something larger at play in his machinations. In total, Axel's depiction in CoM is openly coded as misdirection: it's plain as day that we aren't seeing this person for who he really is, we're only grabbing glimpses of the cards he elects to flash us from his hand. This is how Axel presents himself when he is managing a situation wherein he's comfortably in the driver's seat, sewing the kind of chaos only he can make sense of.
Advancing through the series as intended, KH2 deliberately eschews this reading of Axel's personality (in keeping with its thematic purpose of dividing every character into seemingly irreconcilable dualities that nevertheless reveal the truths of their internal realities) by immediately stranding him in a situation where he has
no control. This is indicated textually by placing him in his first appearance inside of a virtual reality containment system that is designed and administered by a separate power, so he's rather literally out of his element, and it only goes downhill from there. In KH2, we see what Axel's like when he continually finds himself forced into a corner: he's no longer the master manipulator, coolly containing unfortunate complications with the backing of a powerful organization, but a rogue on the lam running on fumes and throwing every backup plan he has at the problem in front of him until all his errors finally rebound on him and he's forced to make that final, self-destructive gamble. In KH2, Axel is caught up in a chaotic scenario that is not of his making, and we see the toll that takes on him-- but we still don't have a strong sense of why he made the choices he did, and the ambiguous nature of his arc is summed up when Sora asks him rather directly, "What were you trying to do?" and all Axel can respond with is a fumbling attempt to categorize feelings he himself doesn't fully understand. The problem for Axel in KH2 is not that he suddenly became a different person, but that he was impacted by experiences that confused his fundamental perception of himself: more than anything, Axel was seeking a resolution to his own existential dilemma, and the only way he could find it was by taking himself to the threshold of existential annihilation and, ultimately, crossing it. There's an idea embedded in there that Axel was acting decisively
not in his best interests during the events of CoM, that with every soul he eliminated he was actively digging his own grave deeper, and the increasing uncertainty with which he carries out his actions as an adversary with too many obligations to conflict closing in on him drives that home. Would he have eliminated Roxas during the prologue of KH2 if he hadn't been overpowered, in order to save his own skin? There's no correct answer, because by that point Axel already doesn't know himself well enough to be sure.
Then we have Days, which has as its sole task with regards to Axel bridging the seeming disparity in his depictions. It does this by removing the circumstantial framework dictating both of his previous appearances and allowing Axel to participate in the story not only as himself, but as someone being forced to look at himself for the first time in too long. Thematically, this runs as a tangent with and in many ways an intersection of Xion's journey of self-discovery, where she learns who she is and how to accept it, and Roxas's journey of self-insulation, where he refuses to look at the truth of who he is for fear of what he will see. Axel's age relative to the other two lends this process a unique quality, because it burdens him with a history with which he has to contend, textually referenced as "memories" which neither Roxas nor Xion carry inside their hearts. The question of
why Axel latches onto Roxas, in particular, is a provocative one because it is so central to what occurs within and around him, and while the answer isn't made immediately apparent it becomes gradually excavated as the character drama that defines Days unfolds. First, we have to consider what we see of Axel and how it coheres with what we already know: factoring in his interactions with Saix, these conversations run him as close to CoM as Axel gets outside of actively inhabiting that role, and it reframes his decisions and actions in Castle Oblivion as being done in service of another's aspirations-- in other words, as indicated in KH2, Axel was
not acting for self-serving gain in CoM, and in fact Days goes farther to reason that he's essentially taking on all the risk and getting his hands dirty for a cause which he would appear to regard as greater than himself (sound familiar? we'll get to that). His wits and careful strategic management of the game afoot are the only things keeping his head off the chopping block, so right away the throughline from his introduction in CoM to his demise in KH2 is laid out, but there's still some connective tissue to weave in. It's also in Days that we see the emergence (or re-emergence) of a side of Axel that he had been consciously or unconsciously suppressing in order to act in his capacity as the sword within Saix's agenda: it should be remembered that Axel was lead to believe he had no heart, and thus to distance himself from any experience of emotion as fleeting, erroneous, imagined, and inauthentic-- basically to dissociate from his internal state as much as humanly, or inhumanly, possible, so that the "cold blooded murderer" we see in CoM is a reflection of how successfully he was able to lean into that bifurcated construction of his experience of self as a "Nobody." Suffice to say, for Axel,
being Axel is fundamentally a different thing from being Lea: he can do things as Axel that Lea would never do, and the life that he lived as Lea is like a "past life" that has no bearing on his present intentions. So when Saix refers to him using his old name, it's a way of cutting him down by dredging up the vulnerable, human past Saix knows intimately and through which he can extend his influence over Axel: it's a possessive, emotionally manipulative gesture, although it's rooted in the genuine legacy of their friendship (as all the most potent psychological buttons are in KH).
What we see as Days progresses is the increasing strain in Axel's conflicting attempts to maintain and balance
both aspects of his entire self, which he has never been made to reconcile up to that point. While Roxas and Xion are busy rebounding off of each other in the text, the subtext to Axel's arc is that he is becoming increasingly internally compromised, and that's manifested in the stark divisions put to his loyalties in realtime: he's challenged to make choices that have no optimal outcome, betraying one friend for another, finding his sense of duty to the cause to which he had committed to as Axel shaken by the rediscovery of a moral compass and a personal directive he didn't realize he still carried with him from when he was Lea. And yet, though he doesn't quite get to the point of consciously realizing it, it was that moral directive that had always fueled Axel's intentions: at his heart, Axel is driven by an inverted form of the kind of heroism that is embodied by Sora (I believe this is even alluded to in CoM, with Axel saying something to the effect that they're more alike than Sora realizes), as both are willing to go to extreme and even self-sacrificial ends in order to protect and be of service to their friends. However, while Axel is more logically deductive, Sora is more emotionally intelligent, so the experience of falling in love iiii mean finding someone who makes him feel whole again at the expense of his safe and ordered separation of his internal identity surprises and unbalances Axel whereas that's the kind of depth of connection that Sora actively pursues.
So what is it that Roxas does to Axel, exactly? Well, first of all, he looks like Ven, which seems kind of quaint until we consider that Ven is a marked reminder for Axel of who he once was, reappearing suddenly and out of nowhere and preserved almost exactly to Axel's memory, as if he stepped right out of a time machine. Even if Axel didn't consciously recall Ven (and I believe KH3's journals make it explicit that he did, but honestly who cares about that game anyway), his heart, or the memories that would come to form the basis of it, would have reacted, and we know what happens in KH when the heart responds to something. Axel would have felt that connection, and been inevitably drawn towards it. But there's something more important that Roxas himself does, or more specifically, doesn't do, at least initially: he doesn't ask anything of Axel. The fact that he's empty and guileless means that he can't burden Axel with the kinds of dictations he receives from everyone else around him: Roxas isn't going to ask him to kill people because it's politically expedient, or beat up teenagers to keep them in line. The part of Axel that is Lea is absolutely attracted to that, an unexpected opportunity to kindle a potential connection that isn't made bloody and cynical by time and circumstance. Of course Axel himself doesn't understand this because he's not in tune with his own resilient humanity, but he's nevertheless compelled by it, and as Roxas grows into himself we see how the bond between them develops. Whereas with Xion, who inadvertently holds up a mirror to him and everybody else, Axel is gradually forced back into the position of prioritizing an intelligible order he believes is "for the best," at an inevitable human cost, which has the dual consequence of degrading the progress he had made in revitalizing his suppressed emotional experiences and, ultimately, poisoning his relationship with Roxas, leading to the fallout wherein he basically makes the decision to defy
everything in a last ditch effort to self-actualize. Again, the heroic trope is carefully inverted to match Axel's conflict of character: he goes to save Xion, guided by his best instincts as Lea and in direct contradiction of what "Axel" believes to be best, but he nevertheless intends to do so by applying the methods of his Nobody persona and that bites him (he fails to understand/doesn't care that Xion has already self-actualized, and can't really be dissuaded from the path she set for herself; Xion later makes the opposite mistake in overestimating Roxas's readiness to accept who he is and what must be done). The existential crisis that Axel faces in KH2 is one of losing what he perceived (but did not fully comprehend) in Roxas as a "second chance," someone who makes him "feel like he has a heart" in the sense that he reminds him of who he used to be and almost seems to offer a redemptive path forward for him. In that spirit, it's fitting that his single-minded pursuit of that redemption actually
does restore him to his human self (as I said, it took crossing that threshold and risking total annihilation, but you could say that was the price on Axel's soul).
As for everything post-KH2, I think there was a strong intention in designing a dramatic confrontation between him, as Lea, and Saix, in conjunction with his continued attempts to restore Roxas and Xion. The problem is that DDD is short on opportunities for deep characterization because it sets itself up with too much plot and doesn't distribute it economically, and then KH3 is...KH3, so while I don't think Axel is as thoroughly trashed a character as some, none of this pretext is ever brought to real resolution or payoff in any satisfying fashion. He asks Kairi to call him Axel as a way of recalling something he recognizes instinctively he must have known from his time as a Nobody, buuuut then he plays no part in bringing Xion or even (lol) Roxas back, so it doesn't matter. He crosses weapons with Saix in theatrical fashion in DDD but has no role in shifting Saix's allegiance in KH3, so it doesn't matter. And don't even get me started on Mystery Girl X. It does not matter.
But yeah. Prior to the games that suck Axel had a pretty good arc going for him, all things considered. I would have loved to see a version of KH3 where his skills as a tactician were put to proper use, maybe working with Namine, Riku and Aqua, hell even Ienzo, to develop actual plans of action in the decisive, fateful, potentially world-ending battle instead of walking into an obvious trap half-cocked and then getting mowed down by Super Saiyan Terra. Idk. He had potential, connections within the Organization (honestly, Isa ought to have been his greatest asset, but that kind of basic dramatic turnabout is beyond KH3's capacity to fathom), and that final showdown between him and Xemnas would have been so much more gratifying if he had actually fucked things up for the bad guys once again.