They all run the same ECU and the same firmware on it, yeah. Teams can control some parameters though. It's to stop any kind of traction control.
The clue is sort of in the name; you make the best car within the "Formula", that's why its called "Formula One". The "Formula" is essentially the set of rules that sets constraints on what you can and can't build. You build the best possible car within the Formula. It's "One" because it's meant to be the pinnacle of single seat racing. It's been roughly this way since WWII.