feat: add serializable session state for distributed deployments (#2111)

[gspeter-max]

Summary

This PR adds serializable session state to enable distributed deployments of the MCP Python SDK. Users can now store session context in external storage (Redis, databases, etc.) and restore it across different server instances.

• SessionState Pydantic model: Serializable snapshot containing session_id, protocol_version, next_request_id, server_capabilities, server_info, and initialized_sent
• ClientSession.get_session_state(): Extracts current session state as a serializable object
• ClientSession.from_session_state(): Creates a new session from previously saved state
• Public API exports: SessionState available via from mcp import SessionState

Changes

• Added src/mcp/shared/session_state.py with SessionState BaseModel
• Modified src/mcp/client/session.py:
  • Added _session_id, _server_info, _initialized_sent attributes
  • Added get_session_state() method
  • Added from_session_state() classmethod
• Added exports in src/mcp/__init__.py and src/mcp/shared/__init__.py
• Comprehensive tests: 19 tests (5 for SessionState, 14 for ClientSession integration)

Test Plan

• All 1130 existing tests pass (no regressions)
• SessionState serialization tests (JSON/dict round-trip)
• get_session_state() extraction test
• from_session_state() restoration test
• Full round-trip integration test
• Type checking passes (pyright: 0 errors)
• 100% branch coverage on session_state.py

Usage Example

from mcp import ClientSession, SessionState

# Save session state
async with ClientSession(...) as session:
    await session.initialize()
    state = session.get_session_state()
    json_str = state.model_dump_json()  # Store in Redis, DB, etc.

# Restore session state
state = SessionState.model_validate_json(json_str)
async with ClientSession.from_session_state(state, ...) as session:
    # Request IDs continue from saved state
    result = await session.send_request(...)

Related

Resolves #2111

[gspeter-max]

@maxisbey @Kludex @felixweinberger - PTAL, this PR adds serializable session state for distributed deployments (#2111). Ready for review!