Using two-dimensional (2D) and three-dimensional (3D) kinetic simulations, we examine the impact of simulation dimensionality on the laser-driven electron acceleration and the emission of collimated $gamma$-ray beams from hollow micro-channel targets. We demonstrate that the dimensionality of the simulations considerably influences the results of electron acceleration and photon generation owing to the variation of laser phase velocity in different geometries. In a 3D simulation with a cylindrical geometry, the acceleration process of electrons terminates early due to the higher phase velocity of the propagating laser fields; in contrast, 2D simulations with planar geometry tend to have prolonged electron acceleration and thus produce much more energetic electrons. The photon beam generated in the 3D setup is found to be more diverged accompanied with a lower conversion efficiency. Our work concludes that the 2D simulation can qualitatively reproduce the features in 3D simulation, but for quantitative evaluations and reliable predictions to facilitate experiment designs, 3D modelling is strongly recommended.
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