If you’re a Mac user who found that MacOS Ventura 13.3 broke the functionality of Continuity features including Universal Control, Sidecar, Universal Clipboard, HandOff, and other Continuity capabilities, you should update to MacOS Ventura 13.3.1 (or later) as soon as possible to fix those problems.
MacOS Ventura 13.3.1 resolves issues with Continuity features, despite this particular resolution not being specified in the versions release notes.
MacOS Ventura 13.3.1 update is available to download now via System Settings > General > Software Update.
Assuming Universal Control and Sidecar were setup and functioning previously, you should find that simply updating to MacOS 13.3.1 will resolve the issues, without having to engage in additional troubleshooting steps.
You should also install iOS/iPadOS 16.4.1 on any iPhone or iPad that you are intending to use with Continuity features on the Mac, though you may find that simply updating MacOS to 13.3.1 is sufficient to get the set of features working again. Nonetheless, the recent software updates for iOS and iPadOS include various bug fixes and security enhancements, and it’s good practice to keep system software up to date on your hardware.
Some users find that using Sidecar in MacOS Ventura can be more cumbersome and challenging to setup than in prior versions of MacOS due abandoning the dedicated preference panel for the feature, and similar frustrations can apply to Universal Control. This also can lead to some difficulty when attempting to troubleshoot issues with either feature, since there are no always available settings available in the newly redesigned System Settings.