How to disable Universal Clipboard
Solution 1:
Universal Clipboard is a part of the Continuity suite of features, specifically a sub-feature of Handoff. The only way to disable it is to disable Handoff entirely.
Mac: Go to System Preferences > General > uncheck Allow Handoff
iPhone: Go to Settings > General > Handoff > uncheck Handoff
Data does not travel to Apple's servers or even over the internet. Devices advertise availability to each other via Bluetooth and data is transmitted over Wi-Fi, but only on the local network.