We determine the optimal method of discriminating and comparing quantum states from a certain class of multimode Gaussian states and their mixtures when arbitrary global Gaussian operations and general Gaussian measurements are allowed. We consider the so-called constant-p ̂ displaced states, which include mixtures of multimode coherent states arbitrarily displaced along a common axis. We first show that no global or local Gaussian transformations or generalized Gaussian measurements can lead to a better discrimination method than simple homodyne measurements applied to each mode separately and classical postprocessing of the results. This result is applied to binary state comparison problems. We show that homodyne measurements, separately performed on each mode, are the best Gaussian measurement for binary state comparison. We further compare the performance of the optimal Gaussian strategy for binary coherent states comparison with these of non-Gaussian strategies using photon detections.