Ironically, the author mentions what you've done here directly and calls it "Narrative Take Over":

"We try to communicate "I understand" and go on to provide a personal anecdote. Our story is usually of something that we think is in the experiential vicinity of what someone is sharing with us. This, however, is not empathy."

The easiest way to be kind and considerate is to listen and attempt to understand rather than making assumptions. Even on the Internet :)