Since upgrading everything (Xamarin Studio for Visual Studio, 184.108.40.2068; Xamarin Studio for Mac, 6.1.1 build 15; Android SDK Manager, API 24 and 25), I can no longer simply select a start-up project and then select the target device.
For example, having recently selected the iOS project and selected a Target device, if I then select a Droid project, the Target device drop-down will show only "Start". It does not list the target devices. If I leave the Droid project selected, close the solution and then re-open it, it will now have a list of Android target devices from which to select. Similarly, if I then switch back to the iOS project, the target drop-down again only shows "Start". I then must close and re-open the solution and it now shows the list of iOS target devices.
Interestingly, this is not an issue for WinPhone projects, only for iOS and Droid projects. The WinPhone projects always show the list of target devices.