We propose and systematize a family of mathematical frameworks for quantifying how far two (small, finite) categories are from being equivalent. Rather than a single canonical notion, multiple natural approaches arise from different perspectives: (A) editing / reconstruction cost (a combinatorial, functorial zigzag edit-distance), (B) nerve → geometric distances (topological---Gromov--Hausdorff / homotopy-type based), (C) enriched / Lawvere distortions (metric / enriched-category viewpoint), and (D) invariant-signature embeddings (vectorized invariants and induced norms). For each approach we give formal definitions (with a focus on (A) for detailed development), basic metric properties, several worked examples (finite discrete categories, a 2-object chain vs discrete, and one-object group-categories), algorithmic remarks, and open problems. The goal is a rigorous, implementable starting point for a quantitative theory of "categorical similarity".