They might be, but to a small extent. What you are actually thinking of is America's Army - that game was made with recruitment in mind and to show how the army works. Subtle propaganda like no matter in which team you are, you are always a US soldier, and the opposite side is a terrorist. You pick up their guns, they shoot like shit and they are kalashnikovs etc. Someone on their team picks one up, it's the same gun it's always been and shoots perfectly fine. You literally had to go through "training" to be able to use a specific weapon, or be a medic. You took tests at the end of a lecture lol.
America’s Army was a game developed by the US military for recruitment, but was only the military’s direct and open leadership of the project that stood out in this. The more common approach is a less visible partnership where military propagandists and censors approve what’s allowed through, and exercise some narrative control to be consistent with the propaganda they’re pushing.