"You're not alone" doesn't help

When a person shares their struggles publicly, a very common type of seemingly empathetic reaction is to tell them they are not alone. There are a lot of people going through similar issues. Your problems are not unique.

I could never understand this. Never have I suddenly started feeling better after realizing that my problem is not unique. I never though my problem is unique in the first place, nor have I wondered whether other people experience similar problems. This whole dimension of thinking is not relevant to the experience. I'm either in pain or not, and if I am, other people's pains have no effect over mine.

Also, it sometimes feels like there is a hidden accusation in "you're not alone". Somehow, it is assumed that the person has been thinking of themselves as unique. Why assume that?