Were Arethusa and Dido designed for the same role?

Did the Arethusa and Dido classes have similar roles in mind when they were designed?

No; Arethusa was a compact general cruiser, while Dido adapted the hull concept into a crowded air-defence cruiser.

Arethusa tested how small the Royal Navy could make an effective general cruiser with six six-inch guns and torpedoes. Dido used a related hull for an air-defence mission and packed in ten dual-purpose guns, creating tighter stability and space constraints. Arethusa's modest size and manoeuvrability suited Mediterranean night fighting and Norwegian coastal waters, while Dido emphasized wartime anti-aircraft fire. The 5.25-inch battery also preserved the political and institutional status of a cruiser gun, whereas a 4.5- or 5-inch weapon risked making the ship look like a large destroyer. The US Navy's simpler choice of a 5-inch dual-purpose gun is judged more practical. A shared hull lineage therefore did not produce a shared operational requirement or armament logic.