Are there modern airplanes with two propellers driven by one single engine?
NASA has proposed to use single gas generator cores to mechanically drive multiple fans in various advanced vehicle concepts, but those are a long way from actually being built. 





Airbus and others are also proposing hybrid-electric airlines where a few gas generator cores drive many small fans, although in that case the power is electrically distributed rather than mechanically linked. There are some hybrid-electric vehicles in development that might qualify (Airbus E-Fan-X, UTC 804).  With all the current development in electric aircraft, it is likely some vehicle that meets your description will fly in the near future, with the caveat that the linkage between the engine and multiple propellers is likely to be electrical and not mechanical. 

Images from NASA