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