Someone hugely overreacts to some offense the person perhaps didn't even commit, but you do want to express sympathy for their hurt feelings: Phrase is totally appropriate. In the link you pointed to, the sister's friend flipped out over something that shouldn't have been an offense in the first place. The only apology she could make was one of sympathy, where she expressed regret that the person was hurt. I still think it has the potential to cause further offense, of course, if the other person can't see reason. But I do think that in this case, it's fine.
Someone is offended over something that is actually offensive: Phrase is not appropriate. You want credit for apologizing for your actions, you don't say, "I'm sorry you feel that way." You say, "I'm sorry I did X." For instance, if the sister in the link had insulted the friend in front of a bunch of people, then stalked off to catch her ride, she should have apologized for her actions. In such a case, apologizing for the other person's hurt feelings is essentially blaming the victim.