Ya'll realize this is just saying a woman doesn't need a man? Like, a man also doesn't need a woman, or a fish a bicycle. Doesn't mean men as a whole aren't important or don't do important things that society needs. It's just that a woman isn't lacking if she's without one (visa versa for the fellas).
Healthy relationships are generally good for everyone involved though, meanwhile fish can't use bicycles. Not a great analogy, just a funny put-down to stop the pestering.
As a woman who used to do cash in hand construction work, I can confidently say more women would do those things if they didn't have to put up with a hostile work environment
At the Guinness brewery, they had a human-sized animatronics fish riding a bicycle, I bought a poster with a cartoon of it when I went. It must be a more common saying in Ireland, it was also in a U2 song.
FWIW this is a pretty old expression, apparently first recorded in print in 1975. Somehow, society has managed to survive its existence for all this time.