[FEATURE] Mobile app: more context to Code chats

### 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

In the Code chats that we have in the mobile app, only the chat name appears. But it could at least show the folder where it is, or at least a tag indicating which custom group it belongs to, in case this chat on the desktop belongs to a custom group. Because, the way it is now, you don’t know which project this session specifically belongs to, and you end up completely lost.

### Proposed Solution

In the Code chats that we have in the mobile app, only the chat name appears. But it could at least show the folder where it is, or at least a tag indicating which custom group it belongs to, in case this chat on the desktop belongs to a custom group. Because, the way it is now, you don’t know which project this session specifically belongs to, and you end up completely lost.

### Alternative Solutions

_No response_

### Priority

Critical - Blocking my work

### Feature Category

CLI commands and flags

### Use Case Example

_No response_

### Additional Context

_No response_

No files, tests, or entry points are named. Start by locating the mobile Code chat list and the data available for each chat, then check whether folder or custom-group context is exposed to the mobile client. Done means chats consistently show enough project or group context to distinguish sessions without changing the chat name.

Written by the indexing model from the issue text.