You should use DependencyService  to get it each platform.
For android 
Android.Provider.Settings.Secure.GetString(Android.App.Application.Context.ContentResolver, Android.Provider.Settings.Secure.AndroidId);
For ios 
you could refer to the blog or optionally save the IdentifierForVendor e.g. in your AppDelegate and return this value in your IOSDevice class (using the name in the blogpost).
use UIDevice.CurrentDevice.IdentifierForVendor.ToString() to get the device ID on iOS.
Update :
from iOS 8, we cannot retrieve the serial number of our iDevice directly,if you want to get it,you could follow this
For Android,you could refer to this