The great thing about Cookies in each of the mobile platforms is they are shared cookie containers across the app. The WebView doesn't hold the Cookie's the platform does and hence even HTTP calls and different WebView's can all access the same shared cookie container.
However they can be tricky to pull out of some platforms, in that reflection is needed. If your SSO solution only requires them going to webpages to sign on first, then they all share the same cookies anyway and you won't need to do any of this.
Android
Android has 2 separate cookie containers for HTTP and WebView's. Its the odd one out. Hence you have
using System.Net.Http;
private static CookieContainer _cookieContainer = new System.Net.CookieContainer();
private static Android.Webkit.CookieManager _cookieManager = Android.Webkit.CookieManager.Instance;
With HTTP requests you do this
HttpClient client = new HttpClient(new HttpClientHandler() { CookieContainer = cookieContainer });
The WebView uses the other one and you can get and set cookies in each container.
iOS
This one is easy, they are all stored in
NSHttpCookieStorage.SharedStorage.Cookies
WinRT
using Windows.Web.Http; //NOT: Microsoft.Net.Http
var filter = new HttpBaseProtocolFilter();
HttpClient client = new HttpClient(filter);
// Use this, while it comes from an instance, its shared across everything.
filter.CookieManager