The vue-router documentation does not address this topic.
Perhaps my expectation that vue-router could handle this illustrates a fundamental misunderstanding of mine of how subdomains operate.
Could this be handled by vue-router, and if so, how, and if not, why not?
Nope, it's not possible.
vue-router uses history.pushState, which doesn't allow you to change the origin. Quote from the docs:
The new URL must be of the same origin as the current URL; otherwise, pushState() will throw an exception.
In other words - vue-router is a client-side router and you can't change the subdomain on the client-side (in the browser) without reloading the page.