The only way the modern-military setting really impacts play, in my experience, is by the restriction and proliferation of feasible props. You can't have CTF without the "flag" being a suitcase nuke and you can't have king of the hill without there being an IED there to "defuse" and you can't have a map that doesn't look like a war-torn cityscape. At the same time, you can't have a weapon called "assault rifle", because you have to have the FNH F2000 and the M4 and the M16 and the SCAR-L and the SCAR-H and the Mk12 SPR and the AK-74 and the AK-47 and the M14 EBR and the H&K G36 and the H&K G36K and the H&K G36C and the H&K MP5/10/7 with all the optional picatinny rail accessories until the whole thing looks like a gun show and then some nerd goes online and says, "dood if u slap the m249 cover shut with the double link off-center taht shit wil jam 4 sure" and your credibility goes out the window and the ghost of Tom Clancy haunts you all the way to the poorhouse.
The seasoning overpowers the ingredients, basically. It's so bound up in thematic dogma that the design and even the basic rules are enslaved to the setting and props. Billions of dollars have been made, of course, so it can undeniably be a successful strategy. As far as videogames go, however, I tend to prefer less realistic settings like Halo or Unreal or Quake or Half-Life or Turok, since you can have an armory consisting of "Assault Rifle", "Sniper Rifle", "Rocket Launcher", "Pistol" and "Shotgun" and then throw in a Cerebral Bore or a Gravity Gun or a Needler. You don't need a full orchestra with five different types of flute in it to play something I can dance to.
But to answer your question, I'd say that the character progression, unlocks and overall grind associated with many successful games is the culprit here. Battlefield 2 was the first time I found myself thinking, "Well, I guess I'd better play another few matches and focus of a skill or role that I don't particularly enjoy so that I can finally get that medal/gun/ribbon for my profile." Now that's everything. You get guys in Gears of War throwing their whole team under the bus so they can get enough chainsaw kills to ring some imaginary bell. GTA V has custom finishes for guns, and the first one--the very first of eight or more--becomes available (for purchase, no less) after you score 100 player kills with that weapon. Do you see guys running around in deathmatches with their favorite pistol trying to get a shiny gold paintjob on it at the expense of their mates? Yes, you do. Do you see guys driving around in freeplay machinegunning total strangers in order to get the blue SAW? Yes, you do. Would these people behave that way if there was no in-game objective toward which sociopathy is the most direct path? Maybe a few would, but I doubt that these guys, by and large, would hobble themselves if they weren't promised some kind of ear-scratching by the game designers.
But that's a crutch for weak or scanty content, isn't it? Sure, the game is drab and unimaginative, but if you do it blindfolded and with one hand tied behind your back, you get a hilarious pot leaf to put on your profile icon! And you can share that to Facebook so all your friends can see how awesome you are!
Am I rambling? Yeah.
In a nutshell, there are two main things making all shooters feel the same, two big trends. The first is the setting, because soldiers are hip and if you can make someone feel like they're a Force Recon Marine or a Navy SEAL, they'll write you a check. The second is the format, and the granularity of unlocks and endless rankings are conducive to the setting, since you don't have to come up with 85 imaginary medals when there's a big list of them right here in the real world, and you don't have to dream up dozens of pretend guns when you can just take pictures of actual gear and accessories and turn it into a dress-up game for commandos. Having that huge library of content allows you to trickle it out like some kind of satisfaction ration, which I have decided to call a satisfration, compelling players to replay and compete endlessly to earn your shiny, digital love.
Anyway, keep it fresh by thinking critically about how games are supposed to make people feel. People will always respond to incremental rewards, which is why slot machines and heroin and paychecks will never really go out of style, but a game that knows its a game should have a crisp interface, intelligible rules and a rewarding pace.