I am in Vancouver, BC and talked to my ICBC agent today. Yes, all cars in BC are insured by ICBC for basic coverage and liability.
But supplemental insurance can be added on with another insurer. That said, the ICBC agent claimed that in the case of classic cars, an agreed to value could be reached with ICBC. Similar to what everyone here was saying Hagerty and others were willing to do. I am not 100% sure I trust her on that, but... ICBC has a history of wanting to write-off anything with damage over blue/black book. Cars that could still be used being written off. Ugh.
Side Note: there was a Hagerty booth at today's Luxury Supercar Weekend (Pagani, McLaren, Jaguar, Aston, classic muscle, classic JDM, classic brit - automotice awesomeness). Unmanned every time I went by, but clearly they operate in BC.
Kind of annoying they were never in the damn booth, so I can talk to them.