[FEATURE] Add a persistent visual identity to the desktop app

### Preflight Checklist

- [x] I have searched [existing requests](https://github.com/anthropics/claude-code/issues?q=is%3Aissue%20label%3Aenhancement) and this feature hasn't been requested yet
- [x] This is a single feature request (not multiple features)

### Problem Statement

When using AI desktop apps from different providers on macOS, I often place each app in a different Space and switch between them throughout the day.

Many AI desktop apps now have very similar interfaces: a sidebar, conversation list, chat area, input box, etc. After switching Spaces, it can be difficult to immediately tell which app I am currently using.

Unlike the Dock or app switcher, the application window itself often lacks a clear and persistent visual identity.

### Proposed Solution

It would be helpful to display a small, recognizable app logo or icon somewhere in the main interface, such as in the sidebar header or another unobtrusive location.

The logo does not need to be prominent. A subtle, persistent visual identifier would be enough to make the app immediately recognizable when switching between macOS Spaces.

This would be especially useful for users who regularly work with AI desktop apps from different providers side by side or across different Spaces.

### Alternative Solutions

_No response_

### Priority

Medium - Would be very helpful

### Feature Category

Other

### Use Case Example

_No response_

### Additional Context

_No response_

The issue names no files, tests, or entry points. First confirm where the macOS desktop interface is implemented and how its main interface is structured; done means a small, persistent visual identifier is visible without being prominent when switching Spaces.

Written by the indexing model from the issue text.