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](https://mathoverflow.net/questions/288909/) or the field of [circle packing](https://en.wikipedia.org/wiki/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](https://en.wikipedia.org/wiki/Y-%CE%94_transform) on certain types of discrete Riemann surfaces, e.g. [isoradial graphs](https://arxiv.org/abs/1012.2955) or ["surfel" surfaces](https://arxiv.org/abs/0802.1617)

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.