Hello eugene1026,
Thank you for reaching out to Apple Support Communities, and we'll be happy to help in any way we can. We understand you're having issues with your default browser working correctly. Try the steps listed here to make sure the settings are correct from the related article above:
"Change the default web browser
Find out which version of macOS you're using, then follow the appropriate steps.
In macOS Ventura
- From the Apple menu in the corner of your screen, choose System Settings.
- Click Desktop & Dock in the sidebar.
- Scroll down and choose a web browser from the “Default web browser” menu on the right."
If the behavior doesn't change, what we would recommend, is testing this issue in another user account. What we need to do, is log out of your current user. Log back in under another Standard or Administrator account and see if the same issue occurs there. This way we can tell if it is an issue with your user account, or if it is system wide.
Keep in mind, Google is an example of a Search Engine. Whereas, Google Chrome is a Web Browser.
We hope this helps.
Cheers!