As motors became popular for driving machines, people could take their treadle only machines and get them "upgraded" with a motor and light. Unless someone did the motor/electrical modifications at home, my understanding is that if the same were done by a non Singer dealer then the badge would have been removed as part of the modifications, so the setup may have been done by a Singer dealer back in the day.
I also am curious to see the back of the machine and how the motor is mounted.