It's striking how long it's been since there were pop culture narratives that announced a new social-cultural generation.
Millennials started with the indie hit Thirteen in 2003, and really thrust themselves into the mainstream with Mean Girls in 2004. Those two set the tone for the rest of the 2000s (Juno, Superbad, etc.), and their influence runs right up through the latest major generation-defining movie, Lady Bird from 2017. That movie is Millennial to the core, starring later Millennials who are portraying earlier Millennials. It is set in 2002, perhaps imagining itself to be a "Millennial movie before it was popular," i.e. before Thirteen or Mean Girls.
That span of time has also seen a proliferation of reality TV portraying Millennials, mainly on MTV. For shows following a social circle over time, it began with Laguna Beach in 2004, which was spun off into The Hills in 2006, and culminated in two series -- Jersey Shore, and 16 and Pregnant, each beginning in 2009.
Rather than a new series of reality shows following a new generation, those original ones are still on the air, only now showing the Millennials not as teenagers but as adults 25 and older. Jersey Shore: Family Vacation portrays 30-somethings rather than early 20-somethings, as does The Hills: New Beginnings. Teen Mom OG has changed the format from showing current teenagers who are mothers, to those who are late 20-somethings but who were teenage mothers a decade ago.
You might offer the younger characters in Stranger Things from 2016 to now (but the other half of the young cast are Millennials in their 20s), or those in the indie movie Eighth Grade from 2018. Maybe in five years we'll look back and see them as the first in a series of Gen-Z narratives. Still, where's the first mainstream movie like Mean Girls or The Breakfast Club?
Until we see something like that, it's premature to refer to Gen Z as a social-cultural generation. They must have a collective self-awareness of their culture being a distinct break with the last generation before them. And so far, just about all narrative "youth" culture is still focusing on the Millennial audience, many of whom are now over 30.
Gen Z may (or may not) be aware of themselves as a distinct group in the technological, economic, or political domains of life, but certainly not as a distinct social-cultural group.
It seems like the first generation-defining movie comes out around the time when a generation's earliest members are 20 years old -- The Graduate (1967) for Boomers (late '40s births), The Breakfast Club (1985) for Gen X (late '60s births), and Mean Girls (2004) for Millennials (mid-'80s births).
This suggests that, given the absence of such a movie by 2019, Gen Z does not include late '90s births, who are more like late Millennials. Then Gen Z begins at least in the early 2000s -- possibly later -- and we won't know until the first defining mainstream movie comes out. Eighth Grade may be the indie prelude, though, so the major hit may arrive sooner than later, which still puts Gen Z as those born sometime during the 2000s.