fix: forward stdio server stderr through errlog

[jordansilly77-stack]

Summary

• pipe stdio server stderr on POSIX instead of handing the current sys.stderr object directly to the subprocess
• forward stderr chunks through the existing errlog stream
• close the stderr pipe during shutdown and add a regression test for errlog forwarding

Fixes #156.

Testing

• python3 -m py_compile src/mcp/client/stdio.py tests/client/test_stdio.py
• uv run ruff check src/mcp/client/stdio.py tests/client/test_stdio.py
• uv run ruff format src/mcp/client/stdio.py tests/client/test_stdio.py --check

I could not complete the local pytest run on this Mac because the repository virtualenv hits local plugin/import issues, including macOS blocking black's compiled extension and slow logfire/pydantic plugin imports. The focused code and formatting checks above pass; CI should provide the full pytest signal.

AI disclosure

AI assistance was used to research the issue context, draft the patch, and run local checks. The final changes were reviewed before submission.

[maxisbey]

Thanks for the PR. Tracking this in #3268 instead. Closing this as part of a wider backlog cleanup following the v2 launch. Feel free to reopen if this is still relevant.

_{AI Disclaimer}