What video game is Billy Batson playing in Shazam! Fury of the Gods?

I will venture an answer, incorporating a few of the observations made in the comments.

Analysis:

• The graphics are ..off:
  • The animations of the hostages are rigid and unnatural. The central hostage doesn't seem to move at all. For a game that has an end screen dedicated to "Hostages Killed", I would assume some effort would go into making these hostages more sympathizable.
  • In that same vein, the faces of the hostages are awkward, low-definition, and have a style that is too cartoony for the rest of the setting (and almost manga-like).
    Their costumes are also low-definition and badly UVed.
  • The models of the hostages have inconsistent definition, as well. Their shoes are early noughties at best, whereas their helmets are far more detailed. Their knee-pads consist of texture only, and have no polygonal definition at all.
  • The ruin of the building in the background, on the other hand, is far more intricate, which doesn't make sense from a performance point of view.
  • Directly in the background is a very low-res skybox of what I think is supposed to look like a desert hill with vegetation, but it doesn't blend with the rest of the scenery at all, and even the scale is off. However, this was not uncommon for games from that time.
  • The skybox material either ends in, or is partially occluded by, a weird structure that doesn't fit in with the rest of the scenery.
  • The normals of the wall to the left have wrong UVs, even different from those of the diffuse texture. (It also seems like this noisy normal is repeated across all the walls throughout the scene.) For a focal point of a game (this is where the hostages are), this seems like a glaring mistake.
  • The lighting is surprisingly good. The over-the-top bloom is common for mid- to late noughties games. Real-time shadows, like that of the flashlight, was gaining popularity, too, during that period. It's impossible to tell what kind of lighting the rest of the scene uses, though. However, what is really surprising, is the use of (the relatively expensive (and AFAIK at that time non-existent)) penumbra (see the lower left corner here). But maybe the shadows on static meshes are all baked.
  • The low-poly ropes seem to have small transparent particles, making them appear woolly. Why?
    All of those meshes are the exact same, by the way, even when slightly rotated for a varied effect, as in the case of the tied-up soldier to the left.
  • The double rainbow is a nice touch; quite redeeming :)
• Apart from a reticle, there is no HUD. This is highly uncommon for a military video game, especially since the player is holding a gun.
• The timer of the explosive is off. It goes from 1:17 to boom in a little over one second. Maybe it simply counts seconds and milliseconds, but that seems unlikely.
• The player who is responsible for blowing up the hostages keeps playing intensely throughout the "Mission Failed" screen.
  The other player, pointing out how this genre of games "is practice for our other job", leisurely does the same, ruling out the idea of them playing in turn.
  This screams the director telling them to "pretend you're playing a video game". Of course, the scenes could have been filmed separately and edited together, or the actors could have been playing an entirely different game altogether, but it is remarkable nonetheless.
• The single-screen, single-gun game is played by two people, simultaneously. How does this work?

Conclusion:

I very much doubt this is a(n existing) game, but if it is, it was made by people who seemingly rarely play (this type of) video games. Kudos to the person who insisted on having a double rainbow, though.