Using the word fishbowl in this context is a metaphor. A fish in a fishbowl has the world staring at it; from the perspective of the fish, nearly anyone can stare at it with curiousity, but it cannot really reciprocate, and has no place to retreat.
The term fishbowl is often applied that way to describe the feelings of celebrities, who may feel very little privacy, even when going to run a small errand – as in, "I feel like a fish in a fishbowl."
A fishbowl moment, then, is a discrete time when someone becomes keenly aware that the whole world (or maybe just the whole neighborhood) is watching them with interest, observing and critiquing even petty details, such as what outfit the person happens to be wearing.