fix: prevent process hanging in stdio_client context manager

[hwaxxer]

Motivation and Context

Fixes#360

How Has This Been Tested?

Tested minimal example:

import asyncio
from mcp import ClientSession, StdioServerParameters
from mcp.client.stdio import stdio_client
async def run():
server_params = StdioServerParameters(
command="claude",
args=["mcp", "serve"],
)
async with stdio_client(server_params) as (read, write): async with ClientSession(read, write) as _:
pass
print("exit ClientSession")
print("exit stdio_client")
if __name__ == "__main__":
asyncio.run(run())

Breaking Changes

No

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

[mroch]

#372 made the same fix

[ihrpr]

duplicate #372