Definitely not just flagship but all campuses of a given university (but not necessarily a "system", where that structure is utilized for overall governance that overarches separate, independently administered universities).
Also, all of these numbers include graduate school alumni so schools that have had traditionally large enrollments in masters and PhD/doctoral programs, medical, business, and law schools also add in to these alumni totals. Even as a private school NYU has had sizable grad-level enrollments for many years of its history. Harvard too, for example. Perhaps smart business decisions as schools can make a lot of money from higher tuition graduate programs.
NYU is by no means a small school even though it's private....it even had a high undergrad enrollment in the 20-25K range going back to the 90s IIRC. About as big as Rutgers-NB undergrad at that time. But my guess is that one of the primary reasons for its climb up this particular list is the more recent development about 3-4 years ago of its full acquisition of Polytechnic University (formerly Polytechnic Institute of NY) in Brooklyn, which is now known as NYU's Tandon School of Engineering. Polytechnic also churned out lots of grads over the years as it is one of the older institutions of higher education in the city, so that alumni pool is now entirely counted by NYU. I believe Polytechnic was founded as the 2nd oldest engineering school in the country, behind only RPI. By the way, university-wide NYU has an enrollment these days in the 50K range so they'll keep churning out alumni at a pretty decent clip.
But similarly, Rutgers' recent acquisition of most of UMDNJ's academic units brought in a sizable bump in the number of alumni from that school over its existence from the past 45 or so years, now counted by Rutgers.
