There are various ways of putting additional data on a fixed 2-complex and then defining a notion of "discrete conformal equivalence" between different assignments of data, see e.g. the answers to this question or the field of circle packing.
There are also notions in these settings which allow you to change the combinatorics of the 2-complex; e.g. one can subdivide triangles in a circle packing or perform star-triangle transformations on certain types of discrete Riemann surfaces, e.g. isoradial graphs or "surfel" surfaces
This is a rather broad field with influences from conformal geometry, combinatorics, statistical physics, and computer graphics and the references given above are by no means meant to be complete or even representative.