[Identity] Improve MI error messages (#43231)

* [Identity] Improve MI error messages

For synchronous ManagedIdentityCredential, if a token request is
unsuccessful, MSAL will return a dict containing "error" and
"error_description" keys. We should ensure that the "error_description"
value is in the raised exception.

Signed-off-by: Paul Van Eck <[email protected]>

* Update sdk/identity/azure-identity/azure/identity/_internal/msal_managed_identity_client.py

Co-authored-by: Copilot <[email protected]>

---------

Signed-off-by: Paul Van Eck <[email protected]>
Co-authored-by: Copilot <[email protected]>

Author: pvaneck

sdk/identity/azure-identity/CHANGELOG.md

@@ -10,6 +10,8 @@
 
 ### Other Changes
 
+- Improved error messages from `ManagedIdentityCredential` to include the full error response from managed identity endpoints for better troubleshooting. ([#43231](https://github.com/Azure/azure-sdk-for-python/pull/43231))
+
 ## 1.25.0 (2025-09-11)
 
 ### Features Added

sdk/identity/azure-identity/azure/identity/_internal/msal_managed_identity_client.py

@@ -61,7 +61,7 @@ def _request_token(self, *scopes: str, **kwargs: Any) -> AccessTokenInfo:  # pyl
             )
         error_desc = ""
         if result and "error" in result:
-            error_desc = cast(str, result["error"])
+            error_desc = f"Token request error: ({result['error']}) {result.get('error_description', '')}"
         error_message = self.get_unavailable_message(error_desc)
         raise CredentialUnavailableError(error_message)