extension: support multiple simultaneous sessions

[davidfant]

What

Enable multiple simultaneous MCP relay sessions in the Chrome extension so different MCP clients can attach to different tabs at the same time.

Why

The extension previously assumed a single active connection, which made parallel agent workflows impossible (or caused one session to kick the other). This change makes parallel sessions viable and improves UX by avoiding tab/window focus stealing.

Tests (packages/extension/tests/extension.spec.ts)

• Launches a browser with the extension
• Opens two distinct pages/tabs
• Starts two MCP clients (parallel-A, parallel-B)
• Connects each client to a different tab via the extension connect UI
• Verifies isolation by checking that each client’s browser_snapshot returns the expected page content, including after running both snapshots in parallel

Notes / Compatibility

• Backward compatible with existing UI consumers via connectedTabId (legacy field).
• Multi-session support is implemented primarily in the extension layer; the test is explicitly labeled a local spike to validate end-state behavior.

How to test

1. Start two MCP clients using --extension
2. Connect each to a different tab from the connect UI
3. Confirm snapshots/actions remain isolated per client/tab

[davidfant]

@microsoft-github-policy-service agree