It could, but keep in mind many OBD-II ports are out of reach. The one shown in the marketing is real, but many are located on the passenger side or a lot deeper on the driver's side.
OBD-II might be a standard, but as I said it was designed to be used while the car is in a garage, never while it is being driven. So there is no "standard" for manufacturers to position it.
I bet it could. But how would it ever turn its self back on? I mean if it is receiving OBD-II information then it knows when the car is running, but if it isn't (which it wouldn't be if it is "off") then it couldn't know to switch its self on.
The phrase you need to google for is "battery isolator". Unfortunately most are rated at like 40 amps for RVs and you want a 40 milliamp one. Many ham radio guys know all about little boxes that connect when the voltage rises above 13.7 volts and disconnect when the voltage applied drops below 13 volts or so. Traditionally you don't use a microprocessor for this, its a beefy FET and two resistors in a voltage divider and not much else. That can be kinda drifty and imprecise depending on resistor and FET tolerance.
Another way is the car squirts ODB-II "stuff" at the port when its on, so little more than a diode/cap/FET combined properly will more or less work.
The strangest way I ever saw relied on what amounts to "hearing" alternator whine over the DC bus. If the alternator isn't spinning while the engine's running, you've got issues. Any source of KHz range noise would activate them, including unfortunately high powered CB radios.
If the device has its own battery power, it could probably use a transistor to just disconnect from the battery voltage completely and go into a low power mode where it just checks for ODB2 information periodically, or even just turns itself on when your phone is in range.
One reason to own a scan tool is to debug anti-lock brake sensors by driving around and noticing which wheel always reports zero... then replace that one sensor. This is far superior to random shotgun troubleshooting. Also while I had a thermostat problem I drove around quite awhile with my scan tool watching the thermostat misregulate my cooling system which is what led to failed readiness indicators because the engine never warmed up all the way only 99% of the way, etc etc. ODB-II is very old, like a college freshman, whereas this is a new iphone app. The equivalent for android, a program called Torque, is old. There are almost certainly HN posters younger than the standard ODB-II port. I bought a bluetooth ODB-II reader for my android/torque phone probably two years ago? It does apparently everything this new app does. My now antique orange physical scan tool appliance that cost me about $150 is about ten years old and obsolete, although it does still work. It is much tougher construction that my phone, which cost a lot more than $150...
One of the nicest Torque features is being able to reconfigure the gauge display. No point in being distracted by the oil pressure gauge when you're driving around if all you care about is a graph of the coolant temp vs time or peak coolant temp or whatever it was.
OBD-II might be a standard, but as I said it was designed to be used while the car is in a garage, never while it is being driven. So there is no "standard" for manufacturers to position it.