So I know this quote is mostly used for Leia, but it doesn't quite fit, Leia had no familial relationship with Vader to speak of, neither of them knew, anything that tackles their relationship is after he's dead, and I honestly think there's way more into it and complicated feelings.
Now, I think many don't even know from where the quote even originated, and while quotes and poems and things can be used outside the context to apply to something different and recontextualize it to make more sense of it, in this case I THINK is very relevant re: Anakin & Palpatine and it's all on the "I'm his favorite-"
The quote comes from Kate from You, here's another quote from her:
"Your past doesn't scare me. It can't be worse than mine."
Kate is the daughter of one of the most powerful men in the world, he owns a business empire, and like she said, she's his favorite, because she's brilliant, a great strategist, and she's fiercely loyal. She had more siblings, but her father wanted her to inherit the business empire, he considered her a prodigy.
Kate is aloof, cold, unsociable, and feels extreme guilt as well, and for a good reason. She grew in that family and she partook in her father's work. She was the one in charge of a sketchy business diverting toxic waste/contaminated water that ended with a bunch of kids dying or with cáncer, which was what made her want run away and change her name.
Here's another quote from Kate:
When I was young, my mother forbade me from crying. If I skinned my knee. Fuck, when my grandparents died. She wouldn't look at me unless I was perfectly stoic. Hide all feeling, and it just became default
Now, in context of Anakin. He is a prodigy and he is a favorite thing of Palpatine, I have spoke of it before, but Palpatine didn't really need Anakin, he just wanted him as a perfect 'fuck you' and an extra score to rub on the Jedi 's face. Palpatine views Vader as his monster, he does care for him, like a prized golden child. This is pretty much how many abusive parents view their children, a score, something they own, something that Represents them; Anakin as a kid was starved for attention, love and to prove himself, Palpatine took advantage of this to groom him and mold him to his liking.
Kate ultimately decided to run away, and start a new life doing something she likes, however her father keeps trying to make her return -she IS his favorite, after all-, she thinks she has made something for herself, but then her father reveals he has been puppetering everything around her the whole time, the realization breaks her, she never really escaped him, in his twisted worldview he helped her, but is just a giant mouse trap, when all she wanted is independence.
Similarly, Palpatine not only positioned himself as someone that Anakin can trust, he also positioned himself as someone powerful, in control of the republic and the galaxy. When Anakin gains 'a seat' in the council, is because of Palpatine's doing and they -rightfully- suspect of Anakin's motives. When Palpatine talks about the Jedi in the opera scene, Anakin rects by trying to defend the Jedi. When Palpatine first reveals he's the Sith lord, Anakin's first reaction is to draw his lightsaber; what I'm trying to say is that Anakin does push back, he finds comfort on being someone's favorite, he finds comfort in being someone Palpatine cares for, there is some privilege in being the favorite, but this is in conflict with wanting freedom and being seen as a person, he pushes back, he tries, but then he gives up if that means the only person he feels on equal footing and that loves him as a person, is going to die, and Palpatine, who already seems to have so much control over everything, offers him a way. And he makes the wrong choice, he makes the worst choice of his life, and regrets it for the rest of his life too.
However, for the rest of his life he remains as Palpatine's favored pet, he is the general of the imperial army, and he his the guard dog. And that's the only thing he has left because from his view Palpatine really literally controls everything.
In legends Vader tried to attempt against his life and Palpatine wouldn't let him. At some point Vader is probably paranoid of Sheev being able to even read his thoughts. Vader doesn't even have control over his damn suit, he doesn't even know for what some things are.
He stopped eating once and all Palpatine did was to make the med droids add intravenous whateverthehell chemicals to keep him going. He has little control over his own body, but he is still the favorite, and he knows that because he's alive, and keeps trying to prove himself to remain in that place, as evidenced by the many times Palpatine pities him against someone else (inquisitors, assasins, the red guards, a long etc).
So Palpatine is the worst man alive, and Anakin his his favorite daughter monster, and there is guilt implied in being the favorite.