feat(client): store serverInfo from initialization result

[bjoaquinc]

Store server information in ClientSession._client_info after initialization when using the default client info. This allows users to access server details without manually capturing them from the InitializeResult.

Motivation and Context

Resolves #1018

How Has This Been Tested?

Created and ran a live integration test that launches a real MCP server via stdio transport. The test verifies that session._client_info is correctly updated with server info after initialization when using default client info, and that custom client_info is preserved when explicitly provided. Normal server operations (like listing tools, etc...) continue to work correctly after initialization.

Breaking Changes

None. This change only affects the internal _client_info field when using the default client info. Users who explicitly provide their own client_info will see no behavior change. The _client_info field is private and not part of the public API.

Types of changes

• Bug fix (non-breaking change which fixes an issue)
• New feature (non-breaking change which adds functionality)
• Breaking change (fix or feature that would cause existing functionality to change)
• Documentation update

Checklist

• I have read the MCP Documentation
• My code follows the repository's style guidelines
• New and existing tests pass locally
• I have added appropriate error handling
• I have added or updated documentation as needed

Additional context

[bjoaquinc]

Hmmm... the test failure doesnt seem related. Looks like a server failure

[maxisbey]

Please add unit tests for this