As a AmEng speaker, I think of "having a moment" as being a fairly recent idiom, maybe from the past decade or so. My personal classification of it is that it's used by the same generation that uses "bae", "cray", "totes", "YOLO", and terms of similar vintage.

I have no idea if that's true or I'm just out of touch, but that's my impression of the phrase. Basically I feel like I tend to see it in titles of thinkpieces about newly popular things from people who are around 30.