claude-in-chrome: extension stays bound to first signed-in account after logout + re-login

What happens: The Chrome extension appears to cache the session of whichever account first opened it. Signing out of claude.ai entirely and signing in as a different account does not rebind the extension. /login in the CLI completes successfully and reports the same account, but the extension remains on the original session.

Symptom from the CLI side: list_connected_browsers returns [] and every mcp__claude-in-chrome__* call fails with:

Browser extension is not connected. Please ensure the Claude browser extension is
installed and running (https://claude.ai/chrome), and that you are logged into
claude.ai with the same account as Claude Code.

This occurs even when the CLI, the extension, and Claude Desktop have all been confirmed to be on one account.

Steps to reproduce:

1. Open the Claude browser extension while signed into claude.ai as account A.
2. Sign out of claude.ai completely. Sign in as account B.
3. Run /login in Claude Code and authenticate as account B. It reports success.
4. Attempt any browser automation tool call from the CLI.

Expected: Logging out and back in as a different account should rebind the extension, or the extension should surface which account it is actually bound to so the mismatch is diagnosable.

Actual: The extension stays bound to account A. list_connected_browsers returns an empty array, giving no indication that an account mismatch is the cause.

Impact: Browser automation is unusable, and the error message misdirects — it names the exact condition the user has already satisfied, which sends you looking at install state and permissions rather than at session binding.

Environment:

• macOS (Darwin 25.5.0)
• Brave (Chromium-based)
• Claude Code CLI

Note: Brave is not the supported browser, which may or may not be a factor here. The account-pinning behavior is the primary report; please don't treat the browser choice as the whole explanation without checking the session-binding path.