Just as the drawing of a physical point on a piece of paper has area while the actual Euclidean "point" that it is trying to represent does not, a specific grid point on a piece of graph paper has area whereas the actual taxicab "point" does not. It is not that some sort of error is made to be small by making the grid more fine, rather the coarseness of the grid is simply to aid visualization. In reality, taxicab geometry is continuous, rather than discrete, just as Euclidean geometry is.
Edit: For clarification, the "points" in both the Euclidean and taxicab case can be represented in Cartesian coordinates (e.g. (x, y)). What is different is how you define the distance between those points. In Euclidean geometry, the distance is r = Sqrt((x - x')^2 + (y - y')^2), whereas in taxicab geometry the distance is r = |x - x'| + |y - y'|.
> It is not that some sort of error is made to be small by making the grid more fine, rather the coarseness of the grid is simply to aid visualization.
Indeed, if one were trying to understand, say, the locus of e^(x + y) = x^2 + y^2, which is an unfamiliar shape then I would say by all means to discretise it; that's what visualisation software would do, after all.
However, conic-section analogues defined via linear constraints on distances will, in the taxicab metric, always consist of unions of line segments, and it seems to me that discretisation is likely to hurt, not help, visualisability of such shapes.
Edit: For clarification, the "points" in both the Euclidean and taxicab case can be represented in Cartesian coordinates (e.g. (x, y)). What is different is how you define the distance between those points. In Euclidean geometry, the distance is r = Sqrt((x - x')^2 + (y - y')^2), whereas in taxicab geometry the distance is r = |x - x'| + |y - y'|.