Kratos's home city in the God of War series is Sparta.
Kratos is introduced as a Spartan warrior who becomes a commander and later serves the Greek god Ares. His identity as a Spartan shapes the original trilogy's military imagery, flashbacks, and relationship with the Spartan army.
The series uses Sparta as part of Kratos's backstory rather than as the main setting of the games. Much of the original God of War takes place in and around Athens, while later stories move through locations connected to Greek mythology.
Athens is therefore a nearby but different answer. It is a major city in the first game and is attacked by Ares, but it is not Kratos's birthplace or home city.