Is the stereo supposed to stay on once the ignition key is turned off on a Peugeot 307 year 2004?

On all 307 the radio should cut OFF immediately after ignition is switched OFF.
if radio is switched ON while ignition is OFF it remains ON for a period of time that depends on battery and will switch OFF because of economy mode strategy.

Hi, After a minute or two it will turn off but if it doesn't then there is a wiring problem in radio power socket. I came across the same problem 2 years ago. In normal operation when you turn off the ignition, radio should switch off after a specific time (max. 2 minutes). In peugeot's radio socket plug there are two positive feed wires. One for the memory (which delivers current to keep your radio code and radio memory after turning ignition off) and other live wire for the normal operation ( but cuts power during energy saving mode after you turn ignition off). I can't remember exactly the wire colours but at that time I figured out those wires with the help of multimeter and swaped those wires to solve this issue.

I don't have specifics on that vehicle because it is not sold here in the US but most newer vehicles have a "Delayed exit" feature that keeps those components active until the door is opened.