That is my WAG. The Walkman’s battery compartment may have something in it (a microswitch or third electrical contact) that recognizes a SONY Walkman brand rechargeable battery. This will (1) prevent the Walkman from trying to charge a non-rechargeable battery, and (2) allow SONY to charge lots of money for their special battery. Look in the battery compartment and see if there’s something like it.
If this is correct, there’s no harm in putting a non-Walkman rechargeable AA batteries in it. But you have to take the batteries out and charge them using a separate charger.