I encountered multiple times the issue when the build of Xamarin app for Android simply didn’t work in release mode without any obvious reason. Finally looks like I got what should be the appropriate configuration.
So, make sure that you have “Options” for Android project as described below:
1. Build -> General
2. Build -> Android Build -> Linker -> Linker Options
3. Build -> Android Application (Minimum: 15+; Target: 21+)
Pay attention to the fact that “Target Framework” in the item 1 should be the same as “Target Android version” in the item 3, you can find the table with appropriate relations between Android name and APIs here: https://source.android.com/source/build-numbers.html