Investigations of the ultrafast acoustic response of metal nanosystems yield important information on the validity of continuous elastic mechanics at the nanoscale and also provide an optical way to probe nanoobject morphologies. In this context, we used femtosecond time-resolved pump-probe spectroscopy to study two classes of bimetallic nanoparticles: chemically synthesized AuAg nanospheres in water in the 20-45 nm size range, both with alloyed and segregated core-shell morphologies, and mass-selected glass-embedded PtAu core-shell nanospheres in the very small size range (2.3-2.5 nm), synthesized by physical methods. The analysis of the corresponding breathing mode periods demonstrates validity of the predictions of the continuous elastic model for bimetallic nanoobjects with the investigated sizes, morphologies and composition. Moreover, discrimination of nanoparticles internal structure (alloy or core-shell) by measurement of their acoustic response is also demonstrated.