As my title suggests, I can turn the car off, remove the key, walk away, come back and (without inserting the key) I can power on the radio and even get the climate fan to spin by pushing the air recirculation button. Is that normal?

I forgot to mention, everything works as it should with the radio turning on when the key is switched on and the radio turns off when the key is turned off. Same with the climate controls.
Nothing remains on unless I push the power button on the radio or the recirculate button on the climate controls.