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One or the other must converge, because we're on a curved surface. As we are on a rotating body there is a good reason to align the symmetry of the body as it rotates to the symmetry of the coordinate system we use. This is why we choose the lines of longitude (the meridians) to converge and maintain (local) parallelism between lines of latitude.
If you really wanted, you could arbitrarily define an east and west poles, with converging lines of longitude. It would be a pain in the arse, but you could do it. However, the north and south poles would still be important because they arise from the symmetry.
If you want to talk more about the curved geometry then the maths thread is a better place I suspect.
hamrack
Why do lines of longitude converge but not lines of latitude?