"Public" in this context is the English use. They are private schools, usually early established ones.The vast majority of Victorian players that represent Vic Country and Vic Metro come from one of the schools in the (APS) Associated Public Schools of Victoria or (AGSV) Associated Grammar Schools of Victoria.
APS Schools: Brighton Grammar, Carey Baptist Grammar, Caulfield Grammar, Geelong College, Geelong Grammar, Haileybury, Melbourne Grammar, St Kevin's College, Scotch College, Wesley College and Xavier College.
AGSV Schools: Assumption College, Camberwell Grammar, Ivanhoe Grammar, Marcellin College, Mentone Grammar, Peninsular Grammar, Penleigh and Essendon Grammar, Trinity Grammar and Yarra Valley Grammar
Even the "public" schools are for all intents private schools, easy to get a scholarship if you are good at footy.
The commentators mention them because that's where they send their kids ( what else can you do when you are seriously overpaid?), and because they offer scholarships to good state school kids for the last 2 years. They never mention which school Patrick Dangerfield, Lewis Jetta , Michale Walters, Wil Powell, Cameron Guelfi ..... In WA, a lot of parents send their kids to PSA schools because they have a higher chance of being seen, and representative selection is soooo corrupt-they often don't select more talented state school kids. (e.g. wil Powell).