I can't say the character supposedly expressing sympathy ever said anything close to "I care about your pain". Everything she said was meant to make the victim feel her pain was minimized, not cared about. And, unless we actually feel someone else's pain, we cannot honestly say we care about it. If someone talked to me like that about something that was really bothering me, I'd just turn around and walk off, for I would not feel that person cared in the least about what I was going through. That character's response was very similar to, Oh well, be glad you're not feeling even worse. That is not sympathy.
People express sympathy in different ways.
I have a friend that does what you describe. I believe that they're uncomfortable expressing their sympathies and their way to deal with it is to try to fix, or temper the situation by comparison.