[Feature Request]: Show search suggestions when triggering search in new window

### Guidelines

- [x] I have [searched the issue tracker for open and closed issues](https://github.com/FreeTubeApp/FreeTube/issues?q=is%3Aissue+sort%3Arelevance-desc) that are similar to the feature request I want to file, without success.
- [x] I have searched the [documentation](https://docs.freetubeapp.io/) for information that matches the description of the feature request I want to file, without success.
- [x] This issue contains only one feature request.

### Problem Description

1. Search for something
2. Click on the searchbar
3. See that search suggestions appear

https://github.com/user-attachments/assets/d73e0931-c641-4966-9689-41af6cea7608

1. Search for something while opening the search in a new window, SHIFT + Enter or SHIFT + Click on Search button
2. Click on the searchbar
3. See that no search suggestions appear

https://github.com/user-attachments/assets/7909ab84-f166-4918-864c-ef9b937bf433

4. Go to original window
5. See that search suggestions appear

https://github.com/user-attachments/assets/86ebb43b-decf-4605-8d2e-194311a28068

### Proposed Solution

Trigger suggestions search in the new window when the search bar receives text from the other windows

### Alternatives Considered

None

### Issue Labels

improvement to existing feature, ease of use improvement

### Additional Information

_No response_