It is, genuinely, in the Geneva Conventions that nobody should use the red cross except for to designate medical staff and establishments that are protected under the conventions. The idea is to make sure that there is absolutely never any doubt that that symbol means anything else in order to minimise the risk to those people

The red cross sign has a very specific meaning and protection under international law. They don't want the symbol to be used outside of the agreed uses because they don't want that meaning, and consequently the protection it affords, to get muddled.

They also don't like when a red cross is used on a random first aid kit in the real world.

I think they were worried about their branding being associated with violence. In the 90s, there was a ton of anti-video game propaganda branding it as violent garbage that was corrupting our children.

Ironically, video games are very good at conditioning human responses to iconography. So despite the Red Cross’ hostility, video games still succeeded in conditioning a lot of people to instantly associate plus signs (any color) with health.

Its pretty obvious to me.

The red cross, red crescent and red crystal emblems provide protection for military medical services and relief workers in armed conflicts.

Under no circumstance is it acceptable to create a situation where something could be mistakenly identified as being associated with the Red Cross. If it appears in the game it might appear on a publicly visible computer screen, poster, TV, etc and thats not acceptable.

i think it has to do with trademarking law
if enough people use is to mean other stuff they can lose their trademark

edit: i looked it up and its called genericize