Not every discussion about important topics is a structured logical argument in which you can point out fallacies. Instead (imho) it is better to analyze these types of situations in terms of the rhetorical/persuasive technique(s) being applied.
From Aristotle's Rhetoric
Of the modes of persuasion furnished by the spoken word there are
three kinds. The first kind depends on the personal character of the
speaker; the second on putting the audience into a certain frame of
mind; the third on the proof, or apparent proof, provided by the words
of the speech itself.
If Alice says "I used to think X, like you do, but now I think Y" to Bob, this by itself addresses the first two modes of persuasion. By acknowledging that Bob's position at least seems reasonable sometimes, Alice is generating goodwill between herself and her audience. Said in a slightly different way, since everybody thinks they're reasonable, Alice is saying "look I'm reasonable too -- I shared this belief with you (a belief that you current think is still reasonable)." Carried along with this is an implicit idea that Alice is knowledgeable about the reasons why you might come to believe X; another boost to Alice's character and a facet of similarity on which to base goodwill. Finally, by calling out the idea that people can, and do, change their minds about X you've introduced this idea into the audience's frame of mind.
The third mode, apparent proof, isn't directly manifest in your short description, but if Alice were to walk Bob through where she used to stand, and why/how she's changed her mind can provide a compelling case (in the very least it was compelling to Alice herself). Thus arguing from "I used to stand where you do" and explaining how and why you've changed is an effective rhetorical technique that addresses all of the modes of persuasion.
Often, this is a form of, or relies on, the ganfalloon technique of persuasion.
If all that your friend said "I used to think like you do" and left it at that, then you can see why it falls down -- it doesn't address the third mode of persuasion at all.