The implementation of the Share feature is intended not to work here. The sample project you are referencing is a simple proof of concept that the UWP SDK also works for Xamarin projects. The project description also mentions: " It shows a simple PDFViewCtrl with annotation functionality as well as a more advanced viewer (iOS and Android only)."
The UWP side of the Xamarin sample does not fully implement the Viewer as there are complete samples already implementing these same features, such as the CompleteReader sample project.
As for implementing the sharing feature, we use the Windows DataTransferManager, more information can be read here. Implementation example can be found in the CompleteReader sample project.