I too like it, in fact I find myself missing it in other languages. I will say that anonymous interfaces are kind of rare in Go, and more generally interfaces in my experience as used in real codebases seem to lean more on the side of being producer-defined rather than consumer-defined.
But there are a couple of problems I can think of.
The first is over-scoping: a structural interface is matched by everything that appears to implement it. This can complicate refactoring when you just want to focus on those types that implement a "specialized" form of the interface. So, the standard library's Stringer interface (with a single `String() string` method) is indistinguishable from my codebase's MyStringer interface with the exact same method. A type can't say it implements MyStringer but not Stringer. The solution for this is dummy methods (add another method `MyStringer()` that does nothing) and/or mangled names (change the only method to `MyString() string` instead of `String() string`) but you do have to plan ahead for this.
The second is under-matching: you might have intended to implement an interface only to realize too late that you didn't match its signature exactly. Now you may have a bunch of types that can't be found as implementations of your interface even though they were meant to be. If you had explicit interface implementation, this wouldn't be a problem, as the compiler would have complained early on. However, this too has a solution: you can statically assert an interface is supposed to be implemented with `var _ InterfaceType = ImplementingType{}` (with the right-hand side adjusted as needed to match the type in question).