Building a framework with shared code on Android and iOS using React Native allows integrating React Native into both a native mobile app and a shared library. The shared library can contain React Native code and be published as an npm module. To integrate it, the library initializes its own ReactNative instance with a separate root view. This allows running two independent ReactNative instances in the same app. There are advantages like code reuse and fewer bugs, but also challenges due to React Native's immaturity and differences from native development.