I'm getting a user's inbox via /me/inbox. I am forced to specify a site with this, so I'm setting it to Stack Overflow. However, it still gets all my notifications, including those from other sites. What is the point of having to specify a site if it makes no difference in the results?
You should use /inbox instead, which is a network route instead of a site-specific one. As mentioned in the documentation, the site-specific route is mostly just provided for convenience of consumers working in a single-site context.
Even then it doesn't seem particularly useful, but meh. To answer your question in regards to why site is required there, it's because /me routes just substitute in the authed user ID for the request's user ID parameter, and that requires a site (since it's not the global account ID). This makes more sense for the other /me routes than it does this one, of course.