[Feature Request]: Chat Sync Across Devices

### Problem Description

I often use your chat application on both my PC and mobile phone. However, I've noticed that the chats I initiate on one device do not appear on the other. This lack of synchronization between devices makes it difficult to maintain the context of conversations when switching devices.

### Solution Description

I propose the introduction of a chat synchronization feature across devices. This would allow users to start a chat on one device and seamlessly continue it on another.

### Alternatives Considered

_No response_

### Additional Context

I believe that this feature would be highly beneficial to many users and would significantly improve the overall user experience.

No files, tests, or entry points are named in the issue. Start by mapping how chats are stored and associated with users and devices across the web and mobile clients, then define the synchronization scope and acceptance criteria before implementation.

Written by the indexing model from the issue text.