[FEATURE] Show native browser context menu on second right click

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

Claude Code in browser and desktop app completely overrides the native context menu. This is an anti-pattern in 99% of cases. Users want the native browser context menu, because there is no amount of features you can fit in the your own menu that can replace the native one (even if you try; the native menu may have items from extensions etc.).

Proposed Solution

• First click — Claude Code's context menu appears.
• Second click — Claude Code's context menu disappears, and the native browser context menu appears.

This is how other web apps (e.g. Telegram) do if they really want their own menu.

Alternative Solutions

• Show the app's own menu when there is NO text selected. Show the native menu when there IS text selected. (This is how Discord does it.)
• Just drop the app's own context menu and use other UI channels.

Priority

High - Significant impact on productivity

Feature Category

Other

Use Case Example

Plenty of those scattered across the issues section. Users ask for various context menu options:

• #61484
• #66131
• #61004
• etc.

What they most likely want in reality is just the native menu.

To the list above, I'd add the ability to open a Claude Code dialog in a new tab, because there is no such option now:

<img width="397" height="338" alt="Image" src="https://github.com/user-attachments/assets/914f4310-58b4-4324-b7dc-69324ea930f3" />

This should not be solved by adding a new option (and then one more, and then one more...). This should be solved by returning access to the native context menu.

Additional Context

_No response_