Do termites like synthetic stucco? What is it made of? Sawdust?

No, it's the wood framing underneath the synthetic stucco that can't really be easily monitored or checked that's the problem.

I'm not familiar with the termite threat, but water damage issues with EIFS (the technical name for synthetic stucco is Exterior Insulation and Finish System) is a big and famous deal: https://en.wikipedia.org/wiki/Leaky_condo_crisis

The basic issue is that you can't really check what's going inside the EIFS so if water (or termites, I suppose) does get inside it can't get out and you can't really know until it collapses because the wood framing is gone. Even if you check every year, if the probe missed the bad spot then you've no way of knowing.

For such a wet damp environment like British Columbia what is the recommended or tried and true way to build without suffering rot?

I just searched online and homeowners insurance policies typically don't cover termite damage.

I assume this isn't actually a huge financial risk since mortgage companies don't care if you have it? In other words you'd self insure for it.

In most parts of the country it's not a very big risk. Most insurance doesn't cover earthquake damage either by default, though it can often be added.

Mine does cover as long I use a termite service for prevention. But wouldn't for this house.