Is that not because the copper holds more heat, so stays hot for longer at the same dissipation?

Dissipation = thermal conductivity. Copper is better in both, it's just heavier and far more expensive. Are you sure you put the same amount of energy into both blocks there? A copper heatsink can generally be much smaller than an aluminium one.

The problems and complications with your method:

1. Suppose Cu is much faster at heat transfer and you only briefly touched it. Your skin does not actually sense temperature, but rate of heat transfer, which depends not only upon the material temperature but also upon how well it will transfer heat. Better use a thermometer.
2. As explained in https://lemmy.nz/comment/20463232, you need to make sure both have been given the same amount of thermal energy and not just heated upto the same temperature. The best way to do so, is by embedding a heat generator right in the middle of the block and transferring a measured amount of energy. e.g. You can embed an electric heater wire (you will require insulation too. That's your headache (ceramic, perhaps)) and pass current using a power source that gives a measurement of the total energy.