Unfortunately for technical reasons snap areas are currently bound to a specific display. You can have a different set of snap areas for your other display.
In the future I plan something to transfer the between displays.
Yep, different IDs and potentially different positioning relative to the main display in System Preferences => Display.
It should keep them for both displays separately, but if they both report the same serial number (unfortunately some screens do this) this might not work. You could try whether setting the exact same position in System Preferences on both systems works, if everything matches (serial number + positioning), they should show up on both systems.