It it possible to have the cigarette lighter socket always on for charging phones?
 

When I remove the key from the ignition, the charging port turns off too. I can't find anything in the manual for how to make it remain active. I'd like to be able to charge my phone without the car being on. Is this possible?

Correct, you must have the key in the ignition for the electrical system to be active.

I'm not sure about your particular car but my 2000 jeep has 2 ports a cigarette lighter and a power port. the power port has constant power while the cigarette port has switched power. you can simply add a power port (looks exactly like a cigarette lighter port but a cigarette lighter doesn't work in it. then all you have to do is take constant power off any number of feeds under the dash and there you have it a constant power supply. you can add the port for under 5 bucks and in 15 minutes. the only scary part is drilling the whole in the dash.