Religion is a term that is today largely filled with meaning through our more modern experiences with monotheistic religions and secular governments. Especially in pre-Christian times, there was no real division between society and religion. Society was structured and supported by the religious beliefs of groups largely because religion at the time was not really a belief; instead, it was essentially a form of knowledge. Pre-Christian Scandinavians didn't just believe in the land-wights and the gods; they knew they existed in the sense that it was treated as unquestioned truth. Social and political structures were built around these beliefs. Importantly, changes in political and social systems generally led to changes in the religious traditions, which were largely oral and more malleable than we are used to.
The same would have been true in Greece. Athenians treated the existence of the gods and other entities as unquestioned truth. Things like hospitality, justice, and social interactions were suffused with religious structures that elevated the importance of some actions while largely prohibiting others in a social manner. Additionally, these ideals informed the views of Athenians concerning themselves, which also impacted their political systems. For Athenians the democracy they engaged in rested on social structures that dictated who was a citizen, who was free, and how decisions should be made.
Therefore, by examining Athenian religion and democracy, we can gain an understanding of supporting and interconnected cultural elements that suffused the Athenian state.