On one hand, a library can allow its users access to typed structs—this is the way gocraft/web does it. The upside here is huge: applications are now type-safe, and can lean on the compiler to enforce their own correctness. The library code that backs this behavior can be arbitrarily gnarly, but you only have to write it once, after which it'll Probably Be Correct (tm).
On the other hand, you can provide a standard "universal" context type like Goji does. Since the library makes no statements about the sorts of fields you have or their types, applications have to do a bunch of shenanigans with type casts in order to extract the typed values they expect. The upside here, however, is the standardization and what that allows. I've found that in most of the web applications I've built, only a small fraction of the middleware is authored by me: I end up pulling in the standard request logging middleware, or the middleware that assigns request IDs, or the middleware that does CSRF protection. There's a huge amount of value in being able to write (read: "let someone else write") those middlewares once and being able to use them in any application. With arbitrary application-provided structs, this isn't possible [1], but if you can instead standardize on types and keys out-of-band, you can mix and match arbitrary middleware with impunity.
This alone is one of those awkward engineering tradeoffs where you're exchanging one nice thing (application-level type safety) for another (standard interface, and hopefully an ecosystem of drop-in middleware), and given just this I'd argue that the ecosystem is probably more important. But the frosting on this particular cake is that, with only marginally more awkwardness, you can actually get application-level type safety too: look at what I did for the request ID middleware that ships with Goji, for instance (https://github.com/zenazn/goji/blob/master/web/middleware/re...). Again, it's not quite as nice as a user-defined struct, but in the grand scheme of tradeoffs and sacrifices, I don't think it's so terribly bad.
So TL;DR, Goji chose standardization and composability in the design of its context object over application type safety, but if you're willing to write a few helper functions I don't think you end up losing the type safety anyways.
[1]: actually, come to think of it, with a lot of reflection and some struct tags you could probably pull it off. (Something like "struct { RequestID string `context:"reqid"`}", and the request ID assigning middleware would hunt around for a string field tagged "reqid"). But it seems like it'd introduce a lot of coupling between middleware and the application which I'm not sure will turn out well—any time a middleware needed a bit of extra storage, every application would have to change. In any event, if someone builds this I'd love to play around with it.