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Okay, I browsed through tropes, and couldn't find anything, so I'm pretty sure this trope specifically doesn't exist.
However, I'm pretty sure there is (or could be, if it doesn't exist yet) a broader trope here that all these examples are a part of.
I'd like to apologize for all this.
Competitive Multiplayer, variation on Control the Points form.
That's probably it, though it's listed as an FPS-exclusive mode, which is weird. Maybe it's just that the page needs cleanup.
Some Fall Guys minigames like Hoarders also this, now that I think of it.
I'd like to apologize for all this.
A multiplayer mode in a video game based around trying to cover the most of an arena with paint within a certain time limit, with players painting over each other's paint to try and get territory back. Examples of this include the Turf Wars from Splatoon, the Reversi Battle (aka Blocks Battle) from the Bomberman series, and the Blob Party mode from de Blob and its sequel.