"arc of prismatic colors formed by the refraction of light rays by drops of rain or vapor," Middle English rein-bowe, from Old English renboga; see rain (n.) + bow (n.). Common Germanic compound (Old Frisian reinboga, Old Norse regnbogi, Swedish regenbåge, Dutch regenboog, German Regenbogen). The American rainbow trout (1876) is so called for its resplendent colors. Old English also had scurboga "shower-bow."
"a vertebrate which has gills and fins adapting it for living in the water," Old English fisc "fish," from Proto-Germanic *fiskaz (source also of Old Saxon, Old Frisian, Old High German fisc, Old Norse fiskr, Middle Dutch visc, Dutch vis, German Fisch, Gothic fisks), perhaps from PIE root *pisk- "a fish." But Boutkan on phonetic grounds thinks it might be a northwestern Europe substratum word.
Popularly, since Old English, "any animal that lives entirely in the water," hence shellfish, starfish (an early 15c. manuscript has fishes bestiales for "water animals other than fishes"). The plural is fishes, but in a collective sense, or in reference to fish meat as food, the singular fish generally serves for a plural. In reference to the constellation Pisces from late 14c.
Fish (n.) for "person" is from 1750 with a faintly dismissive sense; earlier it was used in reference to a person considered desirable to "catch" (1722). Figurative sense of fish out of water "person in an unfamiliar and awkward situation" attested by 1610s (a fisshe out of the see in the same sense is from mid-15c.). To drink like a fish is from 1744. To have other fish to fry "other objects which invite or require attention" is from 1650s. Fish-eye as a type of lens is from 1961. Fish-and-chips is from 1876; fish-fingers from 1962.
1892, gefüllte Fisch, not a species but a loaf made from various kinds of ground fish and other ingredients; the first word is Yiddish, from German gefüllte "stuffed," from füllen "to fill" (from PIE root *pele- (1) "to fill").