How would you react if I were to make an exact clone of Call of Duty, then call it an Adventure Game because the game has a sense of run and gun adventure! Probably not well because Adventure games as a genre means something else.

I imagine most people wouldn't react at all, because in reality nobody really cares that much about genre definitions to argue about it. Besides games describe their genres all by themselves - if it looks like an FPS, plays like an FPS, then it probably is an FPS, despite what the store page might say.

Breath of the Wild is an Open World, Action Adventure game despite Nintendo's insistence that's it's an "Open Air" game, and Death Stranding is a Open World Adventure game with a hint of Stealth and Survival Horror despite Kojima's insistence that it's a "Strand" game, whatever that means. It really doesn't matter in the end, everyone knows what they are just by looking at them.

Games can be a mix of genres as well - that FPS game you're defining as a Adventure game could potentially be an adventure game in some respects depending on how it plays, while also being an FPS game at the same time. They're not contradictory terms.

Genres are less defined categories and rather terms used to describe the web of relationships and influences that a piece of media has to others.