Two simple reasons.
Different battery chemistries have different voltage and safety.
Standard alkaline batteries are nominally rated for 1.5V, but they can have more than 1.6V when brand new and are considered to be discharged at around 1.1V or so, depending on application.
Standard rechargeable batteries that are intended to be used in place of standard alkaline batteries in consumer electronics have less voltage. Nominally they are rated for 1.2V, for example NiCd batteries can have around 1.4V when fully charged and are considered to be discharged at around 1.0V.
So basically, the device is designed to use 1.5V alkaline batteries, and when using NiCd rechargeable batteries, the device can complain about batteries being empty and shut itself down, even though batteries are not empty yet. In case of NiCd batteries, they degrade faster when only partially charged and recharged, as they perform better when fully discharged before recharging.
And alkaline batteries are safer than rechargeable batteries, especially in the hands of children, as the device is basically a toy.
Rechargeable batteries are able to output roughly 10 times more current than alkaline batteries, so in the event of device malfunction due to mishandling such as dropping it, a short circuit with alkaline batteries is relatively harmless as the wires or batteries may heat up noticeably, but with rechargeable batteries, a short circuit can heat up copper wires red hot, melt the insulation, and start a fire. Not to mention possible hazards when mishandling the batteries outside the unit that might cause them to short, such as keeping spare rechargeable batteries carelessly.
The manual only talks about NiCd batteries, as while NiMH batteries did exist, they might not have been so popular everywhere due to various reasons.