I would imagine putting all the music on the device and just giving each cartridge an address would have been considerably cheaper and easier. This could have been electrically, connecting different pins of the existing battery holder solution; mechanically, such that each cartridge has a key shape that depresses different microswitches in the device; magnetically, using magnets on the cartridges; or optically, using different pattern holes on the cartridges and leds with optical sensors on the device.
I think, personally, I would have gone the mechanical route and just have an array of switches in the device. Then the cartridges can be simple plastic keys and the device can draw no power when there's no cartridge.
I think the Fischer Price record player worked this way: https://www.amazon.co.uk/Fisher-Price-Interactive-Packaging-.... The tracks on the record form a binary number and the record player head has mechanical switches.