"Religion" stems from the Latin "religare". Which means "to bind". "Religio" is a bond, an obligation. Which evolved into "living under monastic vows".
In principle, one can be religious about anything. You can feel obliged to God, the State, the School System, or your mother in law.
"Curiously, for a people so religiously minded, the Greeks had no word for religion itself; the nearest terms were eusebeia (“piety”) and threskeia (“cult”)."
Curiously indeed! Olympus was crowded. Didn't they worship or felt obliged? Was it a cult for them? Piety?
Is religion a modern phenomenon, concept, idea, practice? Is it connected with god(?) uniquely? Like I said, it can be used non-theistically. Wikipedia agrees:
Religion is usually defined as a social-cultural system of designated behaviors and practices, morals, beliefs, worldviews, texts, sanctified places, prophecies, ethics, or organizations, that generally relates humanity to supernatural, transcendental, and spiritual elements; however, there is no scholarly consensus over what precisely constitutes a religion.
From the Online Ethymology Dictionary:
This noun of action was derived by Cicero from relegere "go through again" (in reading or in thought), from re- "again" (see re-) + legere "read" (see lecture (n.)). However, popular etymology among the later ancients (Servius, Lactantius, Augustine) and the interpretation of many modern writers connects it with religare "to bind fast" (see rely), via the notion of "place an obligation on," or "bond between humans and gods." In that case, the re- would be intensive. Another possible origin is religiens "careful," opposite of negligens.
In English, the meaning "particular system of faith in the worship of a divine being or beings" is by c. 1300; the sense of "recognition of and allegiance in manner of life (perceived as justly due) to a higher, unseen power or powers" is from 1530s.
There is more to it... A manner of life, morals, being careful of creation, or a bond with God, it can be associated with religion, but when I think of religion, the first think that pops up in my mind are visions of churches, mosques, rituals, religious culture, power hierarchies (God being just an instrument to maintain it), candles, choirs, Christmas, fastening, Carnevale, saints, the pope, Khomeini, praying, Sundays, temples, offerings, hell and heaven, Olympus, damnation, good and bad, Christ (in Rio), Buddha, Mohammed in Mecca dancing the Hajj , crosses on tops of high mountains, holy books, rising seas, Armageddon, the ark, ten commandments, Bismillah, Beelzebub, boats beneath fire, crusades, missionaries, in the name of... etcetera. And not so much of the nature of God, theodicees, apologetics, the reasons for creation, the meaning of life, or science.