[FEATURE] Support Microsoft Edge as browser connector (native messaging works, feature is Chrome-only)

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

Microsoft Edge is my only browser (no Chrome installed), but the Cowork/claude-in-chrome browser connector only recognizes Google Chrome. When I try to use it, the extension reports "not connected" even though Edge is Chromium-based and runs the same extension fine.

This blocks a real, recurring workflow: I need the browser connector for routine tasks like uploading files to Google Drive, and right now the only way to use it at all is to install a second browser (Chrome) just for this feature.

This was previously raised in #34909 (closed as "not planned") and #24367 (closed as duplicate, registry-only fix). #34909 is now locked, so I can't add this data point there.

Proposed Solution

Recognize Microsoft Edge as a supported browser for the Cowork/claude-in-chrome connector, the same way Chrome is today. #24367 confirmed the native messaging host registers correctly for Edge once the registry entries are added manually — the remaining gap looks like a Chrome-only check in the connector logic itself, not a technical limitation of Edge as a Chromium browser.

Ideally: installing the extension in Edge and connecting it should "just work," without needing manual registry edits.

Alternative Solutions

• Installing Google Chrome solely to use this feature — not viable for an Edge-only setup, and defeats the point of having a single default browser.
• Manually adding the missing Windows registry entries for Edge's native messaging host — even with this workaround in place, the connector reportedly still doesn't recognize Edge (per the discussion in #34909).

Priority

High - Significant impact on productivity

Feature Category

Other

Use Case Example

Example scenario:

1. I'm on a Windows laptop with Microsoft Edge as my only browser (no Chrome installed).
2. I ask Claude Code to upload a file to Google Drive using the browser connector.
3. The extension reports "not connected" because Edge isn't recognized, even though it's installed and active.
4. With Edge support, the connector would work exactly as it does in Chrome, without requiring me to install a second browser just for this one feature.

Additional Context

Related issues: #34909 (feature request, closed "not planned"), #24367 (bug report on missing Edge registry entries, closed as duplicate — now locked).
Native messaging host registration for Edge works correctly when added manually; the blocker appears to be in the connector's browser-detection logic, not Edge itself.