> What would happen if you had to tap a card/phone to get in to the subway system (and this was enforced, no jumping turnstiles), and then have to tap it to get out too.

Not sure about the "measure how long the subway rider has been in the subway system for a continuous period of time" feature, but otherwise that's how subway in Japan works. You gotta tap on your way in and out of the current system you are riding on (as there are multiple competing subway system companies running together even within a given city, often enough with their stops being near each other).

Their reason for doing so is a bit different though. In NYC, your ride is a flat fee, as long as you don't exit subway, no matter where you are going. In Japan, your ride cost is determined by your actual route, as some parts of it have different rates. They actually need to know where you exited in order to calculate the final cost of your ride.