Motivated by mathematical physics, Göttsche and Shende have defined "quantized'' versions of certain enumerative invariants for linear systems on surfaces. Block and Göttsche have proposed formulas for the corresponding multiplicities for tropical curves, refining the classical tropical multiplicities. In certain cases, they were able to prove a correspondence theorem for these refined invariants. In joint work with Sam Payne and Franziska Schroeter, we suggest a geometric interpretation of the Block-Göttsche multiplicities as $\chi_y$ genera of semi-algebraic analytic domains over the field of Puiseux series. In order to define and compute these $\chi_y$ genera, we use the theory of motivic integration developed by Hrushovski and Kazhdan and we explore its connections with tropical geometry.