I am ecstatic to see that you posted this, it's so descriptive without it feeling cluttered with images. The square and alleyways feel alive, have character and texture, and the mystery of seeing this person makes me wonder if it is a stranger or not. It actually reminded me of a scene in the last few minutes of "Your Name Engraved Herein", where two school friends with a lot of history see each other many years later. They've changed a great deal, but still recognize each other, and reminisce on the days when they were together in the square, singing and goofing off.
The separate lines of "I breathe -" held such emphasis, as well as the question of what it means going forward. Were one of you going to meet at the club, or were drawn to it, and now are intrigued by each other? Does the club hold too many memories? Or are you two strangers, wondering what each one, seemingly out of place, is doing there, almost entranced by each other.