The screen capture is from an iPad Pro. It's not clear how it was getting the Macbook clipboard; via AirDrop maybe? Really that just makes it even more concerning.
But you can turn it off by turning off "Handoff", which luckily I already had. Nowhere is it mentioned in the settings UI that this extends to your clipboard.
Handoff really is awesome though. You start browsing a website on your phone, unlock your Mac and boom you can switch to the big screen. Copy a phone number from a website on your Mac, pull out your phone and paste it into the phone app. Once you’re used to it, it seems so obvious
That's great when my threat model is "there's a firm boundary between my devices and everything third-party". Now that that's been punctured, it's a nightmare.