By focusing on YOUR story, you shift the focus away from the customer and his or her situation and emotion. It's actually a common mistake to say too much about one's own experiences with the idea that this will help create customer rapport.
That said, it also depends on how well you know the customer. The more you know him or her, the more leeway you have with relating YOUR experience, but the bottom line is that if you try this, KEEP IT SHORT.
A Better Way To Show Empathy Using Your Own Experiences
Let's say a customer says to you: "I'm so fed up waiting in line here, since I have kids I need to pick up and I'm running late."
You could say: "I know where you are coming from. I just hate it when I get delayed just because there's long lines, and I worry about being late for my kids. They get really anxious if I'm late. Sometimes I don't know what to do"
That's too long.
Better to say: "I have kids too, and it's frustrating being late to pick them up. So, let's see what we can do to get you finished here fast."
This is actually how you combine REFOCUSING with ACKNOWLEDGEMENT, as part of the CARP Model.