No, we don't have a state religion, but our country was originally founded between sections of the country that had French origins, almost totally Catholic, and English origins, almost totally Protestant. The terms of our original constitution contained provisions that required each to fund education in both public and "separate" schools - separate being Catholic in the English provinces and Protestant in the French province. Over the 150 years of our country, these "constitutional" rights to separate, government funded education have be eroding except in my province, Ontario, where the government in the 80's extended additional funding to Catholic schools making them the equivalent of public schools - this party was defeated in the next election, but the funding has remained. Over 30% of the province is Catholic and although about 50% of the province wants to get rid of government funding for Catholic schools, no political party will move in that direction because of the significant disruption to the systems, because the Catholic schools are considered by many superior, and because it would likely be political suicide. Other provinces have moved away from the two systems, but not all. In fact, in Ontario, one political party ran on a platform of funding all forms of religious schools and although they started out as the party in the lead, they ended up being badly defeated in the election. Religious education and elections definitely don't mix.
As an aside, the UN has criticized Canada's education system for funding one set of religious schools but not all. I would be in favor of funding education for all children in all schools who gain accreditation from the government regardless of their faith background.