Answer:
The Iliad and the Odyssey.
Step-by-step explanation:
"The Iliad" and "The Odyssey" were two of the most read and famous epics about Greek culture and civilization. They present and revolve around the heroic acts of Greek kings and the wars they fought. The stories in them contain various aspects of Greek culture, most frequently their belief in hospitality even to strangers, their belief in gods and the divine, and also the presence of gods among mortals.
On the other hand, "The Mahabharata", "Ramayana" and "The Epic of Gilgamesh" aren't from a much different civilization. While the first two are from the Indian civilization, the third is from the Mesopotamian civilization, and therefore, they can't be a part of Greek identity.
Therefore, the verses of the Iliad and the Odyssey were learned by heart and became a part of the everyday Greek identity.