Support cookies in built-in browser

### Feature summary

_No response_

### What problem are you trying to solve?

When iterating on an app locally that needs authentication (for example, via OAuth), it is really annoying that I have to go through the authentication every time I relaunch the browser tab. It would be better if that tab supports cookies so that I can continue to share it with Copilot without the added friction of logging in every time.

### Proposed solution

_No response_

### Workflow impact

_No response_

### Installation context

_No response_

### Additional context

_No response_

Start by locating the built-in browser and tab implementation, then inspect how authentication or session state is currently handled across relaunches. Define the expected cookie lifetime and persistence behavior for local apps using OAuth. Done means a relaunch preserves the relevant login state without requiring authentication again, with coverage for the reported workflow.

索引モデルが issue の本文から書いたものです。