Is car necessary for Athens or surrounding islands?
In Athens absolutely not. Don't rent a car in Athens, the majority of places to visit are easily accessible on foot or by public transport.
In the islands it depends on the island and what you want to do there.
stay in "city" and go shopping or at the beach, hiking, going to isolated beaches, boating, diving, visiting villages...
A car will give you possibilities for travel and visits that you will not have with buses or the few taxis available. Some islands are almost impossible to visit without a car.
What islands?
Impossible to answer for you.
The best known and visited are Crete and 4 or 5 islands among 24 Cyclades islands, it will obviously be these 4 or 5 islands that will be recommended to you. But there are around 220 islands that can be visited in Greece, I have visited almost 20 of them in more than 35 years so I am not going to recommend them all and rather advise you to go to this official and non-commercial website:
https://www.visitgreece.gr/islands/
With 12 days you could stay only in Crete (and you will not be able to see everything) or go to 2 or 3 islands maximum in the same group of islands. Each trip from one island to another takes almost the entire day.
So 12 days can only be a maximum of two 5 days stays (4 nights) on 2 different islands.
Also note that ferries do not go from just any island to just any other. There are some impossible or extremely long ferry journeys