Coupled-wire constructions offer particularly simple and powerful models to capture the essence of strongly correlated topological phases of matter. They often rely on effective theories valid in the low-energy and strong-coupling limits, which impose severe constraints on the physical systems where they could be realized. We investigate the microscopic relevance of a class of coupled-wire models and their possible experimental realization in cold-atom experiments. We connect with earlier results and prove the emergence of fractional quantum Hall states in the limit of strong interwire tunneling. Contrary to previous studies relying on renormalization group arguments, our microscopic approach exposes the connection between coupled-wire constructions and model wave functions in continuum Landau levels. Then, we use exact-diagonalization methods to investigate the appearance of these fractional quantum Hall states in more realistic settings. We examine the parameter regimes where these strongly correlated phases arise, and provide a way to detect their appearance in cold-atom experiments through standard time-of-flight measurements. Motivated by this experimental probe, we finally propose a realization of our model with cold atoms in spin-dependent optical lattices. Our estimates show that the previous fractional quantum Hall phases lie within experimentally accessible parameter regimes, giving a viable route toward their experimental study.