How can I get a signature for a Windows system that will remain unchanged even if the user reinstalls Windows?

The System­Identification runtime class (introduced in the Windows 10 Anniversary Update) gives you a signature for a Windows system that will remain unchanged even if the user reinstalls Windows. There are some caveats, though.

To obtain a value that is consistent across reinstalls of Windows, the method uses the Trusted Platform Module (TPM), or if a TPM is not available, the Unified Extensible Firmware Interface (UEFI). If neither is available, then starting in the Fall Creators Update, the method creates a unique ID and saves it in the registry. The registry value is preserved across upgrades, but is lost if the user performs a clean install of Windows. You can use the Source property to determine how the signature was generated.

The value you receive is specific to the publisher specified in your application manifest. If you are a classic Win32 app with no manifest, then the system will use a generic "publisher" that is used for all publisher-less apps. Signatures generated for apps with the same publisher will match. Signatures generated for apps with different publishers will not match.

Here's some sample code:

// JavaScript

var buffer = Windows.System.Profile.SystemIdentification.
                                      getSystemIdForPublisher();
var id  = buffer.id;
var asHex = Windows.Security.Cryptography.CryptographicBuffer.
                                      encodeToHexString(id);
var asBase64 = Windows.Security.Cryptography.CryptographicBuffer.
                                      encodeToBase64String(id);

// C#

var buffer = Windows.System.Profile.SystemIdentification.
                                      GetSystemIdForPublisher();
var id = buffer.Id;
var asHex = Windows.Security.Cryptography.CryptographicBuffer.
                                      EncodeToHexString(id);
var asBase64 = Windows.Security.Cryptography.CryptographicBuffer.
                                      EncodeToBase64String(id);

// C++/CX

auto buffer = Windows::System::Profile::SystemIdentification::
                                      GetSystemIdForPublisher();
auto id = buffer->Id;
auto asHex = Windows::Security::Cryptography::CryptographicBuffer::
                                      EncodeToHexString(id);
auto asBase64 = Windows::Security::Cryptography::CryptographicBuffer.
                                      EncodeToBase64String(id);
// C++/WinRT

auto buffer = Windows::System::Profile::SystemIdentification::
                                      GetSystemIdForPublisher();
auto id = buffer.Id();
auto asHex = Windows::Security::Cryptography::CryptographicBuffer::
                                      EncodeToHexString(id);
auto asBase64 = Windows::Security::Cryptography::CryptographicBuffer.
                                      EncodeToBase64String(id);

// Raw C++ with WRL

using namespace ABI::Windows::Storage::Streams;
using namespace ABI::Windows::System::Profile;
using namespace Microsoft::WRL;
using namespace Microsoft::WRL::Wrappers;

ComPtr<ISystemIdentificationStatics> systemIdStatics;
RoGetActivationFactory(HStringReference(
    RuntimeClass_Windows_System_Profile_SystemIdentification).Get(),
    IID_PPV_ARGS(&systemIdStatics));

ComPtr<ISystemIdentificationInfo> info;
systemIdStatics->GetSystemIdForPublisher(&info);

ComPtr<IBuffer> id;
info->get_Id(&id);

ComPtr<ICryptographicBufferStatics> cryptoBufferStatics;
RoGetActivationFactory(HStringReference(
    RuntimeClass_Windows_Security_Cryptography_CryptographicBuffer).Get(),
    IID_PPV_ARGS(&cryptoBufferStatics));

HString asHex;
cryptoBufferStatics->EncodeToHexString(id.Get(),
                                       asHex.GetAddressOf());

HString asBase64;
cryptoBufferStatics->EncodeToBase64String(id.Get(),
                                          asBase64.GetAddressOf());

If you want to operate with the raw bytes instead of just encoding them into hex or base64, you can use the IBufferByteAccess interface or the Cryptographic­Buffer.Copy­To­Byte­Array method.