Claude in Chrome: persistent 'account verification' dialog when using Code tab in Claude Desktop

Bug Description

Claude in Chrome browser tools always return "Browser extension is not connected" when used from Claude Code running inside Claude Desktop's "코드" (Code) tab. A "Chrome용 Claude 계정 확인" (Chrome Claude account verification) dialog appears every time, with no way to resolve it.

Environment

• OS: macOS Darwin 24.6.0
• Claude Desktop: v1.1.7714
• Claude Code: v2.1.63 (running inside Desktop's Code tab)
• Chrome Extension: freshly reinstalled from Chrome Web Store
• Chrome Extension ID: fcoeoabgfenejglbffodgkkbkcdhcgfn
• Account: Same account (Max plan) logged in on Claude Desktop, Chrome extension, and claude.ai

Steps to Reproduce

1. Open Claude Desktop app
2. Switch to "코드" (Code) tab
3. Use any mcp__Claude_in_Chrome__* tool (e.g., tabs_context_mcp)
4. Dialog appears: "Chrome 확장 프로그램은 Claude Desktop과 동일한 계정으로 로그인되어 있어야 합니다"
5. All three options ("확장 프로그램 설정 열기", "Chrome 다시 시작", "취소") fail to resolve the issue
6. Tool always returns: "Claude in Chrome is not connected"

What We Verified

• ✅ Same account logged in everywhere (Claude Desktop, Chrome extension, claude.ai)
• ✅ Claude in Chrome toggle enabled in Desktop Connector settings
• ✅ Native messaging host process running: /Applications/Claude.app/Contents/Helpers/chrome-native-host
• ✅ Bridge socket exists: /tmp/claude-mcp-browser-bridge-*/
• ✅ Chrome extension sidebar works independently (shows "How can I help you today?")
• ✅ Extension uninstalled and reinstalled
• ✅ Chrome fully restarted multiple times
• ✅ Extension toggled off/on in chrome://extensions

Relevant Log Errors

From ~/Library/Logs/Claude/claude.ai-web.log:

Uncaught (in promise) UnknownError: Internal error opening backing store for indexedDB.open.

This error repeats multiple times and may indicate a corrupted token/auth store.

Expected Behavior

Chrome browser tools should connect successfully when the same account is used across Claude Desktop, Chrome extension, and claude.ai.

Additional Context

• This has never worked for this user — not a regression
• Related to #20887, #20341, #25736 but those workarounds (changing native host path) don't apply here since Claude Code runs inside Desktop, not as standalone CLI
• The dialog provides no actionable error details, making debugging difficult