There are factory and aftermarket radios that do have a security feature like this. When they are first installed you enter a 3 or 4 digit number using the preset station buttons, and this becomes the security code for the radio. If the power is removed for more than a short period of time, they will not operate unitl this code is re-entered. It is to help prevent theft, since the radio will not work without the code, and if it is stolen, the power will be disconnected. Most times the only way to reset the code is either at the cars dealership, or place of purchase with proof of ownership.
There are aftermarket amplifiers like this also, same function, but the code is entered by way of a type of "morse-code" using a small connector, or there is a 1/8" headphone jack or similar that conects to a small device with a 2-wire serial EEPROM in it. The EEPROM stores a serial number for the equipment, and it is checked on power up, if its not there, the amp won't turn on. This small device is mounted somewhere very inaccessable in the car, hard to find.
If your car uses this security feature in the radio, a dealership is likely the only way to get the code reset.