I can confirm what @warner272 described:
The update to macOS 13.1.1 initially fixed all issues.
Then Handoff / Universal Clipboard stopped working for me again under 13.3.1 for my iPhone.
But this time a restart of the iPhone fixed the issue again (which was not the case before), and in the process a progress window appeared on the Mac for less than a second "connecting to iPhone" (or something like this).
The iPhone shows up in the Finder's sidebar and can be accessed from there. So the iPhone is generally visible to the Mac and part of my setup / Apple ID. But the configuration screen did not appear until I connected by cable. Then I had to "trust the device" again, which I had already done after the updates to the latest versions of iOS and macOS.
I checked the firewall settings as suggested by @VicoMurray: Universal Control is added to allow incoming connections.
For my iPad Handoff still works fine with my Mac, both Universal Control and Universal Clipboard.
On the iPad, too, the configuration screen in the Finder would not show up until I connected via Cable and did the "trust the device" step.