Why is there roman numerals (ii) in copper(ii) chloride, but no (iii) in aluminum chloride?

This is because copper is a transition metals.

Transition metals don't have a definite charge, you can see this on the periodic table. However, Alkali Earth, Alkali, Metalloids, Non-metals, Halogens, Noble Gases, etc, all have a specified charge.

Although, there are exceptions: Zinc, Gold, and Cadmium.

Aluminum is a poor metal, but it has it's own charge (3+).

Therefore, Copper has roman numerals due to the fact that it's a transition metal and Aluminum is a poor metal.

Hope this helps!