The milk bar is an Australian invention. A Greek migrant named Joachim Tavlaridis opened the first in 1932. At that time a lot of Greek migrants were moving to America, as well as Australia, and Joachim visited family in Chicago and saw they were opening shops called soda parlors. So he came back to Sydney with a similar idea, but instead of soda he'd sell milkshakes. At Sydney's Martin Place he opened the first milk bar called the Black and White 4d Milk Bar. And it was a huge hit. So other Greek immigrants copied the idea and within about five years there were thousands around Australia and New Zealand.
Over the years, corner stores and milk bars became the same thing, as they were the go-to business for migrants. So all the Greek and Italian migrants bought the corner stores from the Irish and English immigrants from the previous generation, but started selling all this exotic food from back home. This is why in SA and WA they're called delis.