The Vault Timeout is ignored when the extension is popped out or is opened in the browser's Sidebar.

### Steps To Reproduce

1. Open the Bitwarden browser extension in Chrome and set the Vault Timeout to 1 minute.
2. Pop the extension out to its own window.
3. Wait 1 minute.

### Expected Result

For the Bitwarden browser extension to lock or log out following the set Vault Timeout.

### Actual Result

The Bitwarden browser extension remains unlocked while popped out.

### Screenshots or Videos

N/A.

### Additional Context

The same behaviour is present when the extension is open in Firefox's Sidebar.

### Operating System

Windows

### Operating System Version

Windows 11 Pro 22H2.

### Web Browser

Chrome

### Browser Version

110.0.5481.97.

### Build Version

2023.1.0.

### Issue Tracking Info

- [X] I understand that work is tracked outside of Github. A PR will be linked to this issue should one be opened to address it, but Bitwarden doesn't use fields like "assigned", "milestone", or "project" to track progress.

No source files or tests are named. Start by reproducing the one-minute Vault Timeout in Chrome's popped-out extension window and Firefox's Sidebar, then trace the TypeScript browser-extension timeout handling. Done means the extension locks or logs out in both contexts after the configured timeout, with coverage for each case if the project has relevant tests.

Written by the indexing model from the issue text.