Comment by Moonystic: “It's not just a nickname to him, he NEEDS it to prove that these are 3 separate entities (Angel, Angelus, and Liam). Liam's consciousness (where the personality, preferences, memories, etc. are stored) still exists, it's just that the demon is in control. Otherwise Angel(/Liam) wouldn't be able to reemerge out of Angelus whenever cursed and vice versa.

So Liam's traits were transferred to "Angel," including his lack of mental maturity. That also explains why Angel is acting like a possessive boyfriend. Liam was a womanizer, after all. Except Angel just THINKS he's not Liam, so he allows himself to act superior to Buffy. It's very clear how he feels about her: She's immature. Well, duh!

Anyway, this is him lying to himself that he's this new, made-up entity.”

I’ve chosen to think of the split-divide identities/personalities thing as a mental disorder. It’s a split-personality disorder very reminiscent to Willow/Dark Willow which nobody has any problem understanding as entirely the same person just with a severe trauma complex which can’t be wholly separated or divided from Willow either as well as insanely powerful black magicks she allows to take over so she can achieve her goal of avenging Tara’s death. It’s not a demonic possession of any kind. It’s a coping mechanism to escape/avoid her complex trauma and especially the pain/grief of losing Tara.

Angel/Angelus is no different. The Two Ships is actually One Ship just with being incompatible with time-space boundaries and so it looks as if they’re entirely different people. The supernatural doesn’t work with physics so Angel/Angelus is incompatible with time-space. That’s why it looks like demonic possession but it actually isn’t. And it isn’t for any other significant vampire who mostly retains their psyche as a human. They appear to be the same person just with a lack of a conscience and impulse control on their base desires. Which is more animal than monster to be perfectly honest. There is no other significant vampire in the show that split-divides in identity and personality. Just Angel/Angelus. He is the outlier and inconsistency in the vampire lore. Not everybody else that’s a significant vampire!

Even Vampire Willow is still largely the same person when you consider Willow’s actions and choices later in the show. The sadism, the dominance, the controlling, the power hunger. Of course that would be fully emphasised if Willow lost her soul because it’s deeply repressed with one.

And last but not least - external influence. The condition of external influence. The company a vampire keeps. This has a major affect on how the vampire behaves and perceives. Spike proves this.

Angel/Angelus is the only “demon” that appears to be and behaves like two completely different people but that makes no sense either because they still carry the consciousness of each other. Both consciousnesses are very present in the conscious experience of the other all the time.