It has been wired in wrong.
The new radio will have standard DIN plug for power. That expected a permanent live, and a switched live, plus earth, lighting, speed, an output to electric aerial and earth.
The car will confuse this, by having a permanent live and earth, plus two tiny wires for the data, as the car communicates using Canbus.
The radio is not coping with whatever is connected,so is not giving you the 'On' without ignition.
Correct wiring would include an adaptor, trying to hide somewhere behind the unit, that acts as interpreter between the digital car signals, and analogue expectations of the radio. That would give you the speed signal, if the Nav in the new radio needs it, and a lighting input, to perhaps dim it at night. Fitting is fiddly, and sometimes difficult with limited space. An alternative, mentioned on here often, is to take a switched feed from the 12v socket.