no-load voltage on ac-dc adapter
This is one of the hazards of trying to use a greatly overrated (3:1) unregulated supply. That's actually the first time I've seen a no-load voltage specified on the label, very handy. Yes, it could potentially damage the toy. 

You could try the adapter on the 3V setting and see if the product works, but it might be a bit on the low side. 

Another possibility is to add a dummy load to the universal adapter (something like 10-15 ohms 5-10W), but it might get very hot. Or you could poke around and find a ~300mA 4.5V adapter that can accept your mains voltage.