Unless you’re concerned about your servers ability keep up with four separate games, I would personally just designed to allow for a game identifier.

That way you don't need to duplicate data unless it's called for.

For example you decide Item1 needs to be different for Game3 .. When you change the data for the item is creates a second entry with the new details with an ID for game3 ..


Same goes for user registration, you could simply make one registration but give an identifier for which games they have access to.

Or of course, I didn't understand your question in which case, disregard. But in my experience, centralized has more pluses than minuses.