Wrong.
The specific heat of copper is about 1/3 that of aluminum. So by mass copper stores about 1/3 the thermal energy that aluminum does.
By volume aluminum and copper store about the same thermal energy. This is because copper has about 3X the density.
Not exactly because while by weight copper stores 38.4% of heat that aluminum does, by volume copper stores 142% of the heat energy that Aluminum does. Also these numbers are for pure aluminum which is rarely used in heat sinks. (alloys have lower heat capacities) Also, copper has 169% the thermal conductivity of pure aluminum. Furthermore copper's conductivity is 223% the conductivity of best aluminum alloy for heat conduction and 335% of the worst one. Only silver is better than copper here but not by much (though silver has a lower heat capacity by weight than copper)