iCloud & iCloud Photos · Universal Clipboard and Handoff
Why does text I copy on one Mac appear on my other Mac?
Short answer: That's Apple's Universal Clipboard (Handoff), not Storage Studio. Turn it off on each Mac in System Settings → General → AirDrop & Handoff. Storage Studio's clipboard sharing only runs over Bridge, with its own switch.
That's macOS Universal Clipboard, part of Apple's Handoff — not Storage Studio. When your devices are signed in to the same iCloud account with Handoff on, what you copy on one can be pasted on another nearby.
To turn it off, on each Mac open System Settings → General → AirDrop & Handoff and switch off Allow Handoff between this Mac and your iCloud devices.
Storage Studio's own clipboard sharing only works over Bridge, between devices you've connected yourself, and has its own Sharing switch. Turning that off doesn't affect Universal Clipboard, and turning off Universal Clipboard doesn't affect Bridge. See Bridge devices.