PCのMacアドレスみたいに、iOSデバイスを識別する方法をさがしていたが、
下記の「identifierForVendor」を取得するのがよさそう。

■UIDevice.identifierForVendor
デバイス固有であるが、ベンダごと(Team IDごと)で異なるUUIDを返す。
（転載元）http://nolili.wordpress.com/2012/09/19/uuid/

下記のサイトにidentifierForVendorを取得するソースがあったので、試してみた。
Apple.UtilsはXE4のデモフォルダにあったので、そちらを使った。

■iOS: Identifying Your Users Devices — A Recipe How to Import iOS Classes
http://www.monien.net/ios-identifying-your-users-devices-a-recipe-how-to-import-ios-classes/

■identifierForVendorの取得ソース

unit Unit1;

interface

uses
  System.SysUtils, System.Types, System.UITypes, System.Classes,
  System.Variants,
  DX.Apple.Utils, Apple.Utils,
  FMX.Types, FMX.Controls, FMX.Forms, FMX.Dialogs;

type
  TForm1 = class(TForm)
    procedure FormCreate(Sender: TObject);
  private
    { private 宣言 }
  public
    { public 宣言 }
  end;

var
  Form1: TForm1;

implementation

{$R *.fmx}

procedure TForm1.FormCreate(Sender: TObject);
begin
  ShowMessage(VendorIdentifier);
end;

end.

unit DX.Apple.Utils;

interface

/// <summary>
/// DX.Apple.Utils is a helper library to get easier acces to certain Apple Mac/iOS functions.
/// </summary>
/// <description>
/// This library also imports certain classes or parts of them which have been "forgotten" by EMBT
/// The code is intended to be used with Delphi XE4 only. Other versions may or may not work.
/// There is a dependency to AppleUtils.pas, which ships with XE4 and can usually be founc here:
/// C:\Users\Public\Documents\RAD Studio\11.0\Samples\Delphi\RTL\CrossPlatform Utils
/// Make sure use the most recent version - that samples folder above
/// is connected to an SVN repository of Embarcadero.
/// </description>
/// <remarks>
/// <para>
/// DX.Apple.Utils is Copyright 2013 Olaf Monien / Developer Experts, LLC.
/// </para>
/// <para>
/// All code comes "as is", without any guaranties granted. This code may contain bugs and my not work as advertised.
/// </para>
/// <para>
/// RESTClient is licensed under Mozilla Public License 2.0
///  In other words you are free to use this library in your projects, just leave this header intact.
/// </para>
/// </remarks>

// Logs to the console
// Named NSLog2 to avoid name clash
// iOS device: Xcode - Organizer -> Device - Console
// Mac / iOS Simulator: Mac - Console
procedure NSLog2(const AMessage: string);

// Retrieves the vendor specific device ID - DO NOT USE UIDevice.uniqueIdentifier - this would lead to AppStore rejection since May 1st, 2013!
function VendorIdentifier: string;

implementation

uses
  System.SysUtils,
  Apple.Utils,
{$IFDEF IOS}
  Macapi.ObjectiveC,
  iOSApi.Foundation,
  iOSApi.UIKit,
  iOSApi.QuartzCore,
  iOSApi.CocoaTypes
{$ELSE}
{$IFDEF MACOS}
  Macapi.ObjectiveC,
  Macapi.Foundation
{$ENDIF MACOS}
{$ENDIF IOS}
    ;

{$IFDEF IOS}

// Hack to import forgotten classes/functions and properties
// Be careful - classes with same name may already exist in iOSApi!!
type

  // **** NSUUID

  NSUUIDClass = interface(NSObjectClass)
    ['{D9518F5E-DDBC-4702-A555-411D32B85340}']
  end;

  // We just need  the UUIDString here
  NSUUID = interface(NSObject)
    ['{4C137FF5-E854-461F-B77E-8CD357FD4E9C}']
    function UUIDString: NSString; cdecl;
  end;

  TNSUUIDDX = class(TOCGenericImport<NSUUIDClass, NSUUID>)
  end;

  // **** UIDevice

  UIDeviceClass = interface(NSObjectClass)
    ['{D5105207-FBA7-4F55-BC7B-1ADACE347ECA}']
    { class } function currentDevice: Pointer; cdecl;
  end;

  UIDevice = interface(NSObject)
    ['{481E431F-2C02-4F2D-86C5-7728480ECF48}']
    function identifierForVendor: NSUUID; cdecl;
  end;

  TUIDeviceDX = class(TOCGenericImport<UIDeviceClass, UIDevice>)
  end;

{$ENDIF}

procedure NSLog2(const AMessage: string);
var
  LMessage: NSString;
begin
  LMessage := NSSTR(FormatDateTime('hh:nn:ss,zzz', now) + ' - ' + AMessage);
  iOSApi.Foundation.NSLog(PtrForObject(LMessage));
end;

{$IFDEF IOS}
function currentDevice: DX.Apple.Utils.UIDevice;
begin
  result := TUIDeviceDX.Wrap(TUIDeviceDX.OCClass.currentDevice);
end;

function VendorIdentifier: string;
var
  LDevice: DX.Apple.Utils.UIDevice;
begin
  LDevice := currentDevice;
  result := NSStringToString(LDevice.identifierForVendor.UUIDString);
end;
{$ENDIF}

end.