I am not sure, if you can have multiple games with the same name. If you mean multiple accounts with the same game name, of course. Especially if it is generic names.
Actually, I do not know the requirements to make a game name unique. Probably requires to register a trademark or something like that. And I would guess that should be impossible to get for any generic name.
If you look at movies, there are lots of movies that have the same name. And are not remakes of another.
Usually it is in the best interest of a game (or movie) maker to have a unique name. But there are only so many words. And as soon as you have a theme in your creation, those words tend to sound similar, if you want a short title.