Share a magnetic door contact with a pool alarm and home alarm
I figured the alarms are just checking for continuity. Can that not be done at the same time by two different alarm systems?

No, that is generally not the case.
Besides which, do you really want a fault in one system to affect the other?
Either use two switches on the door, or use a small DPST relay controlled by the switch to multiply the number of independent contacts avaiable.