iCloud & iCloud Photos · Universal Clipboard and Handoff
How do I stop my iPhone sharing its clipboard with my Mac?
Short answer: On the iPhone, open Settings → General → AirPlay & Handoff and turn off Handoff. On the Mac, switch off Allow Handoff in System Settings → General → AirDrop & Handoff. It's Apple's feature, separate from Storage Studio.
Turn off Handoff on the iPhone: open Settings → General → AirPlay & Handoff (wording varies by iOS version) and switch off Handoff. Universal Clipboard stops sharing between that iPhone and your Mac.
To stop it on the Mac side too, open System Settings → General → AirDrop & Handoff and switch off Allow Handoff between this Mac and your iCloud devices.
This is Apple's feature, separate from Storage Studio. If you use Storage Studio's Junk Yard, things pasted across by Universal Clipboard can appear there like anything else you copy. See Universal Clipboard and Junk Yard.