As one can see on Wiktionary, the meaning of the prefix is "co-" as used in mathematics is different from its meaning as used in the rest of the English language, and does not seem to be a natural development from the other meanings.
This leads to my question: when and why did the prefix "co-" first begin to be used in mathematics? Why does is its meaning different in this field than in other contexts?
My guess is as follows: the only place I know "co-" being used in mathematics as in its typical usage elsewhere is in the term "covariant" as in "covariant tensor". This also has a natural interpretation as being the dual object of another important object, leading to other mathematical terms describing dual objects via analogy with covariant tensors, even though the "co-" in the original term with which the analogy was being drawn did not have that meaning.
If it wasn't "covariant tensor", then my other guess would be "cosine". This then would have lead to the name for the "cotangent function", which then would have lead to the name for "cotangent space", which is obviously a dual object and which would have started the trend for naming dual objects "co-". Perhaps the cosine function was called that because people did not feel motivated to come up with an original name for it after translating the Arabic term for "sine".