Dhaust gives a good way to add the property to the ApplicationUser class. Looking at the OP code it appears they may have done this or were on track to do that. The question asks
How can I retrieve the OrganizationId property of the currently logged in user from within a controller? However, OrganizationId is not a property available in User.Identity. Do I need to extend User.Identity to include the OrganizationId property?
Pawel gives a way to add an extension method that requires using statements or adding the namespace to the web.config file.
However, the question asks if you "need to" extend User.Identity to include the new property. There is an alternative way to access the property without extending User.Identity. If you followed Dhaust method you can then use the following code in your controller to access the new property.
ApplicationDbContext db = new ApplicationDbContext();
var manager = new UserManager<ApplicationUser>(new UserStore<ApplicationUser>(db));
var currentUser = manager.FindById(User.Identity.GetUserId());
var myNewProperty = currentUser.OrganizationId;