The question mentions "monotheistic" as in the monotheistic creed-based religions that refer to the Bible or Koran. While these books might refer to old or even ancient religions -- certainly Judaism is ancient by any count -- they are religions of a different sort. As indicated, they are based on a book that includes a set of prescribed practices and beliefs. In contrast, a contemporary example of an ancient religion not based on a specific book and more like the Greek type is Hinduism.
Although there were some atheists among the ancient Greeks, Greek religion pervaded community life. It wasn't a separate sphere. People didn't take breaks each day or once a week to pray to the gods. There was no synagogue/church/mosque of Greece. There were temples, though, to store the statue of the deities, and the temples would be in the sacred spaces (temene) where public rituals would be carried out.
Proper Public Religious Behavior Counted
Personal, privately-held belief wasn't important; public, ritual performance was. While some practitioners of specific mystery cults may have looked to their religion as a way to attain the Afterlife, entrance to Paradise or Hell did not depend on one's religiosity.
Religion dominated most events the ancient Greeks participated in. In Athens, more than half the days of the year were (religious) festivals. The main festivals lent their names to the months. Seemingly (to us) secular events, like athletic festivals (e.g., the Olympics), and theatrical performances were held to honor specific gods. Going to the theater, therefore, combined Greek religion, patriotism, and entertainment.
To understand this, take a look at something similar in modern life: When we sing the national anthem of a country before a sporting event, we honor the national spirit. We, in the U.S., revere the flag as if it were a person and have prescribed rules for how to handle it. The Greeks might have honored their city-state's patron deity with a hymn instead of an anthem. Furthermore, the connection between religion and theater lasted beyond the ancient Greeks and into the Christian era. The names tell it all: miracle, mystery, and morality plays. Even today, around Christmas, many churches produce nativity plays ... not to mention our idol worship of movie stars. Just as the goddess Venus was the Morning/Evening Star, might not the fact that we call them stars suggest deification?
Greeks Honored Many Gods
The Greeks were polytheists.
Honoring one god would not be viewed as offensive to another. Although you wouldn't incur the wrath of one god, by honoring another, you had to remember the first one, too. There are cautionary tales of gods offended that their cults were neglected.
There were many gods and various aspects of them. Each city had its own particular protector. Athens was named for its main goddess, Athena Polias ("Athena of the city"). Athena's temple on the acropolis was called the Parthenon, which means "maiden", because the temple was the place to honor the virgin goddess aspect Athena. The Olympics (named in honor of the home of the gods) featured a temple to Zeus and annual dramatic festivals were held to honor the god of wine, Dionysus.
Festivals As Public Feasts
Greek religion focused on sacrifice and ritual. Priests cut open animals, removed their entrails, burned the appropriate sections for the gods -- who didn't really need the mortal food since they had their own divine nectar and ambrosia -- and served the remaining meat as a festive treat to the people.
Of Central Importance: The Altar
Priestesses poured libations of water, milk, oil, or honey onto a flaming altar. Prayers would be offered for favors or help. The help might be to overcome the wrath of a god angry at an individual or community. Some stories tell of gods offended because they were omitted from a list of gods honored with sacrifice or prayer, while other stories tell of gods offended by humans boasting they were as good as the gods. Such wrath might be demonstrated by the sending of a plague. The offerings were made with the hope and expectation that they would appease the angry god. If the one god wasn't cooperating, another aspect of the same or another god might work better.
Contradictions? No Problem
Stories told about the gods and goddesses, the mythology, changed over time. Early on, Homer and Hesiod wrote accounts of the gods, as later did playwrights and poets. Different cities had their own stories. Unreconciled contradictions didn't discredit the gods. Again, the aspects play a part. One goddess could be both virgin and mother, for instance. Praying to the virgin goddess for help with childlessness would probably not make as much sense or be as propitious as praying to the maternal aspect. One might pray to a virgin goddess for the safety of one's children when one's city was under siege or, more likely, to help in a boar hunt, since the virginal goddess Artemis was associated with the hunt.
Mortals, Demi-Gods, and Gods
Not only did each city have its protector deity, but its ancestral hero(es). These heroes were the half-mortal offspring of one of the gods, usually Zeus. Many also had mortal fathers, as well as the divine one. Greek anthropomorphic gods lived active lives, primarily different from mortal lives in that the gods were deathless. Such stories about the gods and heroes formed part of the history of a community.
"Homer and Hesiod have ascribed to the gods all things that are a shame and a disgrace among mortals, stealing and adulteries and deceiving on one another."