I have a dash cam, too. Chelyabinsk wouldn't have been the same without them.
For a while, I had mine plugged in to the cargo area socket, which is on 24/7. No issues with the car, but I only have ~2.6 hrs of recording time, so it had lots of parking lot footage, and very little driving. I then moved power to the console. Now, the camera chimes when I open the door as if to acknowledge power-on.
It still records in 2 minute files, and has an onboard battery. I don't find partial files, so I suspect that when the console socket goes dead, it runs on the internal battery until that 2-minute segment is done, then shuts down. Working on the car, it won't chime every time I open the door unless enough time's passed.
I suspect yours is staying on using internal power, too. See what happens when you unplug it after you know its on.