What, fundamentally, makes Pick’s theorem possible in 2D that breaks down in higher dimensions?

Pick's theorem allows calculating the area of any 2D polygon (including nonconvex polygons) whose vertices lie on an integer lattice from only the number of lattice points within it and on its boundary. This feels like a minor miracle, and indeed there is no equivalent formula for the volume of polytopes in any higher dimension, even when restricted to convex polytopes. What geometric/topological property of 2D space makes this magic possible that somehow fails in every other dimension?