Not a stupid question at all....
Just a stupid slogan ;-)
Seattle City Light is the name of the company. It is owned by the city, and has been for the last 100 years or so. It really has a fascinating history: An Irish Engineer by the name of J.D. Ross laid claim to the site of two dams, undercutting the local private electric company, which was overcharging the people. He then turned around and deeded them to the city forever, and gets named head of the new public power company. Angry business interests put in a mayor who tries to privatize the company and fires Ross, and the city freaks out and recalls the mayor, and appoints a new mayor (The new guy might have been a socialist, which would have really put the business community in a snit. I know one of the mayors was a socialist, but I can't remember which one. Not that I was there, mind you. I'm not THAT old) Ross gets reappointed the head of City Light until he dies. Sort of a William Mulholland story, but without the St. Francis Dam tragedy.
"Your City Light" is his legacy, because it belongs "to the people". They use the same slogan for all of the city departments, but City Light originated it, and is the most dedicated to it.