In this CBC article from before the Brexit, they discuss what view Queen Elizabeth II might have of Brexit, stating,
But that hasn't stopped some from speculating how the Queen would vote. "As an elderly woman who didn't study at university and lived in a council house, she is in a demographic more likely in which to leave … I doubt we'll ever really know for sure," [pollster Joe] Twyman said. (emphasis added)
Wikipedia describes a "coucil house" as
A council house is a form of public or social housing built by local municipalities in the United Kingdom and Ireland
My understanding is that the royal family lives in Buckingham Palace, is saying she lived in a council house a reference to that, just being underplayed? Wikipedia has very little on her early life, so maybe I just don't know enough about the Queen.