Why are many ancient texts from the Eastern Mediterranean so similar? How do they still display their culture-specific ideology within the narrative framework?

The main reason for the similarities has to do with the high level of contact the eastern Mediterranean societies had with each other. The region is crisscrossed with trade routes that have existed since the Bronze Age. Throughout history, people from one culture interacted with those from different cultures, sharing stories and, later, literature.
Additionally, this region saw frequent conquests and invasions by numerous empires. Many of these civilizations absorbed the peoples that they conquered into their empires. This meant that ideas were further shared, borrowed, appropriated, and adapted.
Since literature is often a reflection of culture, it is no surprise that similarities in writing reflect similarities in the culture of the different civilizations of the eastern Mediterranean. Since many of these peoples belonged to highly organized societies, their writings often include themes pertaining to social order and a group identity that sets them apart from their numerous neighbors.
Even though major themes are similar between the different societies of the region, their texts still strive to set them apart. This is often done by establishing particular customs and laws codified in writing or by alluding to a legendary past with heroes who represent the ideals of their people.

The first reason for their similarities have to do with direct influence. This was a fairly limited geographic area, in which people traded with each other and exchanged stories as well as material goods. Also, rather than nations having fixed and unchangeable boundaries, many of the groups in this area existed in a world of constant conquests and realignment of borders of various empires, with different groups dominating at different times. 
Most of the cultures in this period were extremely patriarchal and socially stratified, something that is reflected in many of their literary texts, which tend to glorify male leadership. Many of the common religious texts and law codes attempt to codify some notion of justice as a principle for social organization that substitutes law for personal retribution. This grows out of the need to organize cities and eventually empires in a way that permits them to function efficiently. 
Although both monotheistic and polytheistic religions existed, they shared in common a manner of expressing religious belief through narrative. The reason for this is that literacy levels were quite low outside a restricted scribal class and thus important cultural information needed to be disseminated in a form that was memorable.