JOEL: Page is ripped out. Don't remember doing that. It appears this is my first entry in two years.
The whiteness of the blank pages in Joel's journal matches the bleak grayness of the sky and the empty beach. A large part of Joel's identity has been stripped away by the procedure, and he is left with a blank slate. Normally the idea of a fresh beginning is portrayed positively, but there's no doubt in this scene that Joel is feeling lost.
CLEMENTINE: Agent Orange! I came up with that one. I apply my personality in a paste.
JOEL: Oh, I doubt that very much.
CLEMENTINE: Well, you don't know me, so... you don't know, do you?
JOEL: Sorry. I was just... I'm trying to be nice.
CLEMENTINE: Yeah. I got it.
Agent Orange was the name of an herbicide used by the U.S. in the Vietnam War. It kills plants; it's definitely not nice. That's why Joel is politely disagreeing—but Clem is having none of it. This is the first time we really get to see their contrasting personalities: Clem admitting her faults and saying exactly what she thinks, while Joel is trying to be kind and undo her self-deprecation.
CLEMENTINE: So, I'm eight... and I have these toys, these dolls. My favorite is this ugly girl doll who I call Clementine. And I keep yelling at her: "You can't be ugly! Be pretty!" It's weird. Like if I can transform her, I would magically change, too.
Even Clem has her own insecurities. As a child she would displace them onto her dolls, having them take on her identity (or at least the parts of it she doesn't like) for her, helping her bear the load. Now, as an adult, she can instead rely on Joel, whose niceness, which at first repulsed Clem, now assuages her self-doubt.
JOEL: Why am I... I don't understand what I'm looking at. Why am I standing here and—
This is Joel's awakening. Well, he's still asleep but you know what we mean: he finally realizes that he's inside his own mind when he sees himself sitting in a chair. Just as Joel is literally looking at himself in this scene, the whole memory sequence is basically one big self-examination, with Joel exploring all of the avenues of his mind, getting acquainted with who he is in a way he probably never expected.
CLEMENTINE: I didn't kill anybody! It's just a f***ing dent, Joel. You're like an old lady or something.
JOEL: Well, what are you like… a wino?
CLEMENTINE: A wino? Jesus, are you from the '50s or something? A wino? Face it, Joely. You're freaked out because I was out late without you, and in your little wormy brain you're trying to figure out, "Did she f*** someone tonight?"
In this fight, Joel and Clem throw names at each other like children, playing on their identities. Joel is boring and always worried like an old lady, while Clem…likes to drink wine? That's probably what wino means, but we're not, like, from the '50s or something.
PATRICK: Baby, what's going on?
CLEMENTINE: I don't know. I don't know! I'm lost. I'm scared. I feel like I'm disappearing.
CLEMENTINE: My skin's coming off! I'm getting old! Nothing makes any sense to me!
PATRICK: You're not getting old.
CLEMENTINE: Nothing makes any sense. Nothing makes any sense.
Clem is lost and scared because she's in the middle of an identity crisis—and not an angst-y teen one or a mid-life one. She's had a real loss of identity, and she doesn't know who she is. She feels older because she really is older than she believes, having had so much time with Joel stripped from her memory. Of course, what Clem is experiencing parallels what Joel is experiencing in his memory. And in Joel's memory, Clem is literally disappearing.