The radiation-pressure interaction between one or more laser fields and a mechanical oscillator gives rise to a wide range of phenomena: From sideband cooling and backaction-evading measurements to ponderomotive and mechanical squeezing to entanglement and motional sideband asymmetry. In many protocols, such as dissipative mechanical squeezing, multiple lasers are utilized, giving rise to periodically driven optomechanical systems. Here we show that in this case Floquet dynamics can arise due to presence of Kerr-type nonlinearities, which are ubiquitous in optomechanical systems. Specifically, employing multiple probe tones, we perform sideband asymmetry measurements, a macroscopic quantum effect, on a silicon optomechanical crystal sideband cooled to 40% ground-state occupation. We show that the Floquet dynamics, resulting from the presence of multiple pump tones, gives rise to an artificially modified motional sideband asymmetry by redistributing thermal and quantum fluctuations among the initially independently scattered thermomechanical sidebands. For pump tones exhibiting large frequency separation, the dynamics is suppressed and accurate quantum noise thermometry demonstrated. We develop a theoretical model based on Floquet theory that accurately describes our observations. The resulting dynamics can be understood as resulting from a synthetic gauge field among the Fourier modes, which is created by the phase lag of the Kerr-type response. This phenomenon has wide-ranging implications for schemes utilizing several pumping tones, as commonly employed in backaction-evading measurements, dissipative optical squeezing, dissipative mechanical squeezing, and quantum noise thermometry. Our observation may equally well be used for optomechanical Floquet engineering, e.g., generation of topological phases of sound by periodic time modulation.