At absolute zero temperature, all substances have the same

Definition of heat capacity: the amount of heat required to raise the temperature of a substance or material by a small amount is called heat capacity.

$$\mathop {\lim }\limits_{dT \to 0} \,dQ = C\,\left( {{\text{heat capacity}}} \right)$$

Every substance has its own molecules and every molecule in a substance has its own properties and structure depending on the nature of the substance so different molecules exhibit different heat capacity at the same temperature. But every substance molecules at 0K will come to rest and to raise the temperature of a substance by a negligible amount doesn’t depend upon the nature of the substance but once the substance reaches $$dT$$ amount of temperature different substance or materials exhibit different heat capacity.