Instead of shading in cells, consider a field of discrete, equally spaced points. For instance, mark the intersections on the graph paper instead of the open spaces.
(Of course, as you say, the error involved can be made 'small', in some sense, by making the grid suitably fine; but at that point, with such a simple metric as the taxicab one, it's not clear to me what you're gaining over just looking at the entire plane all at once.)
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'|.
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.