It can happen. In Boise, Idaho, there is a fuel terminal (big tank farm) where gasoline and diesel is brought in from Salt Lake City, Utah via a pipeline. IIRC, the terminal is owned by Chevron. At the terminal, there are position holders who buy and sell fuel back and forth as demand dictates (remember, all this fuel is still stored in the same tanks). These position holders are usually big name companies, but not always. These position holders then sell to distributors, who may own one or many brands in the area. Or, they may just distribute to independent, unbranded stations.
Since all the fuel is stored in huge tanks, then, yes, all the gas you buy in the area came from the same place. However, the big boys (Chevron, Texaco, Phillips, etc...), usually add their patented products (Techron, Clean System 3, etc...) to the gasoline when they load the fuel on the trucks. The driver will swipe a card and enter a key code. This tells the system to start pumping fuel into the truck. At the same time, octane is set and fuel additives are injected into the fuel as the fuel is pumped through the "rack" to the truck. The additives and fuel continue to mix together as the trucks drive to the station where it will be pumped into the underground tanks.
Thanks for that in depth explanation. It makes perfect sense, and now I know something that I didn’t know before.