It's hard to draw direct comparisons between Vancouver and San Francisco's housing markets. Canada, as well as each of the provinces, local municipalities, and cities, have representational democracy, not a quasi-direct democracy like California.
The spike in affordability in Vancouver wasn't caused directly due to a shortage in housing stock and an influx of residents. It was caused by rampant speculation from people essentially using Vancouver as a safety net to park wealth.
I can't overstate how detrimental this is to a city. What you end up with are entire neighbourhoods which are occupied at 50% or less, and yet with the value of the homes exceeding $2-$4 million. It causes stores in those local neighbourhoods to close because there aren't enough customers. It taxes the fire department because no one is there to call 911 when a fire starts. It guts schools because there aren't enough kids to fill classrooms.
And with all of this going on, former local residents and young people are pushed further and further out of the urban core due to affordability. Vancouverites make less than half of what an average tech salary pays in the SF Bay Area, and the jobs are much harder to come by.
That said, yes, Airbnb and other vacation rentals were exacerbating the problem, and causing Vancouver to turn further into a vacation/tourism destination instead of a real working city. San Francisco suffers from this to a certain extent, however, its real problem is that it has a massive influx of workers due to a booming tech economy, and not enough housing stock to accommodate all of those workers.