Galaxy began as a word for a single smear of light, and it means "milk."
Etymonline traces it to Greek galaxias, short for galaxias kyklos, "milky circle." The root is gala, "milk," which Etymonline links to a reconstructed Proto-Indo-European root for milk. The name described what the eye sees on a dark night: a pale band across the sky.
Late Latin borrowed galaxias for the Milky Way. English then took the word in the late 14th century, either from French galaxie or directly from Late Latin. For centuries it named only that one band. The English term Milky Way is a loan-translation of the Latin via lactea, "road of milk."
The Greeks also told a story to explain the milk. Wikipedia's article on the Milky Way in mythology gives two versions. In one, Zeus lets the infant Heracles nurse from the sleeping Hera, and when she wakes and pushes him away her spurting milk becomes the Milky Way. In another, Heracles bites down while nursing and the milk squirts out. That is myth, but the word really is built from the Greek for milk, as Etymonline and Wikipedia both say.
The meaning of the word changed more than once. Etymonline records a figurative sense, "a brilliant assembly of persons," from the 1580s. The technical astronomical sense, a star system containing the sun and all visible stars, was in use by 1848.
The plural came later. Etymonline says the Milky Way was long thought to be the only galaxy. In the mid-19th century some astronomers suspected that certain spiral nebulae were distant systems like it, but other galaxies were not established until the 1920s.
So the lowercase word is younger than it looks. A term for one pale road of light became a general noun only once people understood that the road was a view from inside a huge disc of stars, and that countless others existed.