Context. The standard interpretation of observed Stokes profiles to infer the physical conditions of the solar atmosphere is inherently an ill-defined problem due to observational noise and mathematical degeneracies. Traditional pixel-by-pixel (1D) inversion codes provide point estimates with unreliable uncertainties, at the expense of significant computational time. Recent machine-learning-based Bayesian frameworks are restricted to 1D spatial configurations, ignoring crucial spatial correlations between neighboring pixels. Aims. We aim to develop a novel multidimensional inversion framework capable of performing fast and scalable Bayesian inference across an entire 2D Field of View (FoV). This approach seeks to provide accurate height-dependent atmospheric parameters with reliable posterior distributions while exploiting spatial correlations. Methods. We introduce a new generative modeling strategy based on conditional flow matching. The model utilizes multi-scale spatial features extracted from observed Stokes profiles in the Fe I line pair at 630 nm, which then conditions a flow-matching generative model to sample from the complex posterior distribution of the atmospheric parameters. The framework was trained using realistic 3D quiet-Sun magnetohydrodynamic simulations. Results. Validation on independent synthetic datasets demonstrates that the model accurately captures the true 3D stratification of all thermodynamic and magnetic parameters. Because the code additionally provides a geometrical height scale, it allows for the computation of 3D electric current density maps, Lorentz forces, and Ohmic and ambipolar dissipation maps in the solar photosphere. Application to real Hinode/SP quiet-Sun observations yields highly localized electric currents at magnetic boundaries. We also leveraged the 3D geometrical information to trace the emergence of small-scale emerging magnetic loops across the solar atmosphere.