'Bint' also means just 'girl'. Actually, I'd imagine most women would find it offensive. 'Bird', for example, meaning 'girl' comes from the same source as 'bride', but most women would be happy to be called someone's 'bride' whereas someone's 'bird' might not be so welcome.
I should have added above that the Arabic for 'woman' is 'hurma', the plural of which is 'hareem'. This is the name often given to a sheikh's collection of ladies or to the place they live in.