Shifts in the excitation energy of the organic chromophore, cis-7-hydroxyquinoline (cis-7HQ), corresponding to the π→π* transition in cis-7HQ and induced by the complexation with a variety of small hydrogen-bonded molecules, obtained with the frozen-density embedding theory (FDET), are compared with the results of the supermolecular equation-of-motion coupled-cluster (EOMCC) calculations with singles, doubles, and non-iterative triples, which provide the reference theoretical data, the supermolecular time-dependent density functional theory (TDDFT) calculations, and experiment. Unlike in the supermolecular EOMCC and TDDFT cases, where each complexation-induced spectral shift is evaluated by performing two separate calculations, one for the complex and another one for the isolated chromophore, the FDET shifts are evaluated as the differences of the excitation energies determined for the same many-electron system, representing the chromophore fragment with two different effective potentials. By considering eight complexes of cis-7HQ with up to three small hydrogen-bonded molecules, it is shown that the spectral shifts resulting from the FDET calculations employing non-relaxed environment densities and their EOMCC reference counterparts are in excellent agreement with one another, whereas the analogous shifts obtained with the supermolecular TDDFT method do not agree with the EOMCC reference data. The average absolute deviation between the complexation-induced shifts, which can be as large, in absolute value, as about 2000 cm-1, obtained using the non-relaxed FDET and supermolecular EOMCC approaches that represent two entirely different computational strategies, is only about 100 cm-1, i.e., on the same order as the accuracy of the EOMCC calculations. This should be contrasted with the supermolecular TDDFT calculations, which produce the excitation energy shifts that differ from those resulting from the reference EOMCC calculations by about 700 cm-1 on average. Among the discussed issues are the choice of the electronic density defining the environment with which the chromophore interacts, which is one of the key components of FDET considerations, the basis set dependence of the FDET, supermolecular TDDFT, and EOMCC results, the usefulness of the monomer vs supermolecular basis expansions in FDET considerations, and the role of approximations that are used to define the exchange-correlation potentials in FDET and supermolecular TDDFT calculations.