"Try selecting that engine and turn it off."
Sadly, it doesn't work. An engine fire is a death sentence in any aircraft in the game. It would be nice if this could be changed for the bombers at least - most other aircraft systems failures can be worked around, except for pilot wounds and severe structural damage. It seems unfair to put the player in an insoluble lethal position. I assume, based on previous posts from Mr HiTech about engine oil damage, that the game makes the aircraft explode after a random length of time has passed rather than anything more complex.
From a programming point of view the creators would either have to add a new set of keys to control the fire extinguishers, or alternatively make it so that if the user shuts down a flaming engine, a puff of white smoke appears, a hissing noise plays, the fire goes out, and the engine becomes permanently broken.
Perhaps the game could be altered so that you could use the cursor keys to move your pilot onto the wing. You would then press the secondary fire button in order to throw a blanket over the flames. And you would have to get back into the cockpit. It would be a nightmare at 20,000 feet.
Your advice will however work in the Me-163. Selecting the engine and turning it off will stop the fire, and also conserve precious fuel, and you can turn it back on again.