Bitwarden extension for Safari doesn't recognize system theme changes between light and dark modes (macOS)

### Steps To Reproduce

1. Launch Safari with Bitwarden extension installed
2. If in light mode, change your system to dark mode (or vice versa)
3. Notice Bitwarden extension UI color theme does not change
4. Fully exit Safari (Cmd+Q)
5. Launch Safari
6. Notice Bitwarden extension now uses correct color theme consistent with macOS theme

### Expected Result

I expect the Bitwarden Safari extension to respect system theme changes and adjust it's own colors accordingly.

### Actual Result

Bitwarden Extension's UI theme does not change after system theme changes

### Screenshots or Videos

### Additional Context

I use Firefox, Chromium (Brave/Edge) and Safari browsers. I only experience this issue with Bitwarden for Safari.

### Operating System

macOS

### Operating System Version

26.2

### Web Browser

Safari

### Browser Version

Version 26.2 (20623.1.14.18.4)

### Environment Versions

Version: 2025.12.0
SDK: 'main (0107af7)'
Server version: 2025.12.1

### 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.

Start by reproducing the theme change in the Bitwarden Safari extension on macOS, switching between light and dark mode without restarting Safari. Trace the extension's theme handling and verify that its UI updates to match the system theme immediately after the change.

Written by the indexing model from the issue text.