Let me try and explain this without going into really advanced cartography. If you take a map of an area, and you draw "square miles" on it, and then you place geographical and human features on it while trying to keep things as accurate as possible: imagine that you don't know that the earth is a sphere-- in this case you will end up with an equirecrangular projection with either a different "standard parallel" than the equator or a different great circle acting as the equator.

And you're right: that's what lat/long lines are for. But you could decide randomly to choose any antipodal points as your poles instead of the axis of rotation: these new points would create new, arbitrary lat/long lines with a new, arbitrary equator.