This also isn't about trademark, the red cross is outlined in the Geneva Conventions as a symbol with a specific meaning. That meaning is specific because it's meant to protect aid workers as neutral parties in conflict.
Good grief. Can you think of a reason why your analogy about changing how a symbol is used in real life might not apply to a debate about how symbols are used in fiction?
Let me offer up an analogy that isn't completely and obviously broken:
If you watch enough Doctor Who, does it make you believe that real life police boxes are actually camouflaged time machines?
The misuse of the red cross, especially in video games, is rampant to the point that the Red Cross is worried about confusion.
There's additional issue with the fact that video games are very common, but warzones and disaster areas less so. So it's quite possible the fictional association, if overused, could redefine the real-life usage for many people. We already see some of this in the comment section... many people don't understand the difference between "health" as a concept and the red cross as an element protected by international law.
It seems reasonable to try and claw that meaning back.
This is the silliest thing I've ever heard, and not every fictional red cross symbol is a threat anyway. This whole thing seems at least as ridiculous as the moral panic about violent video games in the early 2000s, except that I kind of expect ridiculous moral panics from conservative parents not so much from the official communications arm of one of the largest NGOs in the world. Absent any actual evidence I think we're just going to have to agree to disagree.
They're not calling for video games to be banned or even re-labeled, they're just trying to prevent the red cross from losing the intended meaning (which comes with an international treaty intended to protect aid workers)... it seems like the method of applying this mostly consists of asking nicely.