Does anyone know how to prevent this? I'm running Big Sur and hadn't noticed this previously, but I tend to leave a while before updates (precisely because there's always something annoying like this).
When an app appears in the Dock with a badge that looks like a stylized iPhone:
macOS is making you aware that you started something on another device (a Mac, iPhone, iPad, or Apple Watch) and you could pick it up on your Mac. This functionality is called Handoff.
To turn it off:
Launch System Preferences and click General
At the bottom of the pane, deselect “Allow Handoff between this Mac and your iCloud devices”:
More information about Handoff
Apple recently added an article about Handoff to the Mac App Store app: