In this paper we present a detailed analysis of the structural, electronic, and optical properties of an $m$-plane (In,Ga)N/GaN quantum well structure grown by metal organic vapor phase epitaxy. The sample has been structurally characterized by x-ray diffraction, scanning transmission electron microscopy, and 3D atom probe tomography. The optical properties of the sample have been studied by photoluminescence (PL), time-resolved PL spectroscopy, and polarized PL excitation spectroscopy. The PL spectrum consisted of a very broad PL line with a high degree of optical linear polarization. To understand the optical properties we have performed atomistic tight-binding calculations, and based on our initial atom probe tomography data, the model includes the effects of strain and built-in field variations arising from random alloy fluctuations. Furthermore, we included Coulomb effects in the calculations. Our microscopic theoretical description reveals strong hole wave function localization effects due to random alloy fluctuations, resulting in strong variations in ground state energies and consequently the corresponding transition energies. This is consistent with the experimentally observed broad PL peak. Furthermore, when including Coulomb contributions in the calculations we find strong exciton localization effects which explain the form of the PL decay transients. Additionally, the theoretical results confirm the experimentally observed high degree of optical linear polarization. Overall, the theoretical data are in very good agreement with the experimental findings, highlighting the strong impact of the microscopic alloy structure on the optoelectronic properties of these systems.