Majorana zero modes (MZMs) exhibit a fundamental property of non-Abelian statistics, which makes them potentially useful as ideal quantum bits to manipulate certain topological quantum tasks. Rapid theoretical developments have greatly reduced the technological requirements and made it possible to experimentally observe MZMs. However, until now, only a few positive signatures of the formation of MZMs have been reported in solid-state systems, and there has been no investigation of their crucial non-Abelian aspects. Here, we experimentally investigate the non-Abelian character of MZMs in the Kitaev chain model using a photonic quantum simulator. Through the Jordan-Wigner transformation, the geometry phases of the braiding of two MZMs can be directly obtained in an optical system. The local phase-error immunity of MZMs is further studied. This work will be useful for the study of quantum non-Abelian statistics and may provide a novel way to investigate topological quantities of quantum systems.