[Feature Request] ContextBridge - A lightweight cross-platform clipboard sync utility

### Description of the new feature / enhancement

Windows power users who use Android phones currently lack a lightweight, dedicated clipboard bridge. The native "Phone Link" app is excellent, but it is often too heavy and battery-intensive for users who strictly want real-time text and URL syncing without granting full SMS/Photo permissions.
I am proposing a silent, background utility in PowerToys that syncs text/URLs directly between a Windows PC and a companion mobile app.

### Scenario when this would be used?

A developer copies a terminal command or stack trace on their PC and needs to instantly paste it into a mobile browser for testing.

A user copies a tracking link or address on their phone and wants to paste it directly into a spreadsheet on their Windows desktop.

Currently, users resort to emailing themselves or using web-based messaging apps to transfer text. This utility eliminates that friction entirely.

### Supporting information

I have built a working prototype of this concept (called ContextBridge) to prove that the latency is near-zero.

Frontend: Built natively for Windows and Android (Material 3).

Backend/Sync: Currently utilizing Cloud Firestore for real-time WebSocket syncing (though this could easily be adapted to local Wi-Fi peer-to-peer for PowerToys privacy standards).

You can review my working prototype code and system architecture in my public repository here: [https://github.com/SanjitGhose/ContextBridge/tree/main]

Review the PowerToys repository and the linked ContextBridge prototype first. The proposal describes native Windows and Android clients with Cloud Firestore synchronization, but does not identify a PowerToys entry point, tests, implementation boundary, or acceptance criteria. Before work can be considered done, the project would need to define the supported sync architecture and privacy requirements.

Written by the indexing model from the issue text.