"A woman needs a man like a fish needs a bicycle", which is ascribed to Irina Dunn . There is another phrasing in North America "A woman without a man is like a fish witout a bicycle".
I do not believe that this idiom originates from an English speaking country. The phrase makes much more sense in only two languages: Danish and Norwegian. In these two languages, the word for "man" rhymes with the word for "water". In these languages you could say: "A woman without man is like a fish witout water", which is both rythmic and has a rhyme. Some Scandinavian young woman may have heard such an idiom form her mother, and simply have replaced the word "water" with a nonsense word, and chosen "bicycle". As Danes use bicycles much more than Norwegians (Denmark is flat, Norway is mountainous) I would guess that the phrase originates from Denmark. In the years around 1970, some English-speaking feminists may then have heard the phrase from Scandinavian contacts. It would be interesting to find out if the phrase can be traced to Scandinavia already in e.g. 1968 or 1969, i.e. before its appearance in 1970 in the English-speaking world.