All of 3P0's emotional reactions seem to be a bit distant and imitated, if you look closely.

He doesn't really mind telling the stormtroopers that the others went to the prison section (I think?); he kinda accepts serving under Jabba; he readily accepts "new masters" as soon as he's sold to someone else; he doesn't come off as terribly despaired after getting blown apart, and yes, he doesn't seem to be awfully sad about Han, either.
With all that, he obviously shows a lot of personality that could be described as annoyed, concerned, afraid, grumpy, self-centered, compliant, etc.

So why do we care about him? He still shows sympathetic traits (with all the unsympathetic ones being played for laughs), and basically entertains us with his personality (and the fact that it's coming from a robot).
I don't think there's any point in the three films where we truly care for C-3P0 in terms of his well-being or safety - we kinda know he'll get out of this, there is no real threat or fear involved, and he seems to be less involved in the situation than a human would be, too.

How that works? Suspension of disbelief, I guess.

Point is, I find the robot torture scene hilarious :D
Jabba's place is supposed to come off as dangerous and cynical, but at the same time quirky and amusing in dark way.
That torture scene fits that perfectly.

But that's just my perspective.