Here is my understanding based upon fighter lore...

The original Red air F-16's were referred to as "Vipers" to differentiate them from Blue air F-16's, technically the F-16 is the "Fighting Falcon". Over the years the name then took for the entire group, probably for comm brevity reasons; "Kill the Fighting Falcon arcing in a right hand turn at 16K" was too much, whereas "Kill the Viper arcing in a right hand turn at 16K" met the intent of comm brevity!