You can call the studio itself a developer, that is correct. But parts of the whole are not automatically named the same as the whole.
I want to give you an analogy. A car manufacturer. The company is a car manufacturer. The janitor working there is not. The people assembling cars there are on a trivial hair splitting level "car manufaturers", but they would go by their job description.
Or would you call the janitor in a big game studio a game developer? Where do you draw the line? You said, everybody working there... ;-)
And calling a software engineer only by the term engineer sounds wrong, because engineer alone refers to other activities. This is like air taxi and taxi. Both are taxis on a trivial level. But if you say taxi, people understand ground vehicle, not an airplane. If you are talking in established context, you could get away with dropping the qualifier software or air.