I'm also experiencing Safari bookmarks sync problems after upgrading my MacBook Pro to Catalina and iPhone XS to iOS 13.
The issue is: after doing some editing on my bookmarks (adding, deleting, etc.), the changes simply don't sync to the other device.
I've tried rebooting, turning off the firewall in macOS (possible fix mentioned in several threads), deleting all bookmaks on both devices and recreating the structure from scratch, verified Date & Time settings. Even after doing a clean install of macOS and resetting all content and settings on my iPhone, I still have the same problem. Sometimes it may start working for a while, but it always gets to the point where it simply isn't syncing anymore.