You should use [/inbox](https://api.stackexchange.com/docs/inbox) instead, which is a network route instead of a site-specific one. As [mentioned in the documentation](https://api.stackexchange.com/docs/user-inbox), 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](https://api.stackexchange.com/docs/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](https://api.stackexchange.com/docs/me) routes than it does this one, of course.