The iPhone Upgrade Plan is individual. There is no Family package.
Each person involved would need to start the plan, and as such you would need to each pay the monthly fee. Also each person needs to pass Citizen's One credit check to be eligible for the upgrade Program and the phone financing.
Also note, that the upgrade program payment, does not include the phone service which would need to be setup separately with your carrier of choice.
So yes, in essence $140 per month to get 4 iPhones every year. Plus cell service.
Not sure if a single person would be allowed to be be in 4 different Upgrade Programs at the same time however.
See here for specifics on the plan: iPhone Upgrade Program - Apple
FYI: The Trade Up and the iPhone Upgrade Program are 2 different and independent things.