<p>This study tackles the challenge of identifiability in distribution-free anchored linear structural equation models (SEMs), where&#xa0;the observed variables are imperfect measures for&#xa0;the target variables, and the error distributions are not restricted to being Gaussian. It introduces the geometry-faithfulness assumption, ensuring that partial correlations serve as direct indicators of d-separation/connection. The study establishes the identifiability of distribution-free anchored linear SEMs under the same identifiability conditions for anchored Gaussian linear SEMs, but by replacing the faithfulness assumption with the geometry-faithfulness assumption. Moreover, it shows that the learning algorithm leveraging the PC algorithm with Fisher’s z-test, originally designed for anchored Gaussian linear SEMs, remains applicable and effective for distribution-free anchored linear SEMs. It also provides statistical guarantees for the proposed algorithm, including the strong geometry-faithfulness assumption, ensuring its consistency. These theoretical contributions are validated through extensive numerical experiments and the analysis of real galaxy data.</p>