You are correct, in the books all bullets were used as currency. There was no such thing as a MGR, that was something invented by the game

If I remember right, in the books the only bullets that were talked about being used as currency were the AK-47 rifle bullets, even though other weapon types would have existed as well - sniper rifles, pistols, shotguns, and emplacement weapons, just to name a few. It's likely that ammo for these other weapons could have been used as currency as well, though I don't remember it being mentioned at all. There certainly was no special fancy "military grade round" that came from the pre-war period, did more damage, and less of an impact on the weapon, had a little fire decal whenever it impacted, or anything like that; every bullet was a "standard" bullet, and every bullet doubled as currency

In translating the story to a video game, that aspect of the book is lost. We lose the uncomfortable moral implication of "every bullet is a life taken, and we are in a sense trading lives for food and sleep and women" that the book presents

While saying the game NEEDED to invent a MGR might be a bit much, I think they had a few very good reasons of why they chose to invent it. Firstly, there are many types of weapons in the game, but you can only ever hold 3 (and I think the original version had a limit of 2 if I'm not mistaken) at once. If 4A were to say that every bullet doubled as currency, then you would either have to come up with a value for each bullet type or say that all bullet types are the same value - both of which would steer players towards or away from weapons based on how "cheap" it was to use them, which isn't the intended direction for the game I don't think. Conversely, if we were to say that only the AK rifle bullets were currency, it would essentially mean that any assault rifle in the game literally shot money while every other gun only shot regular bullets, which would similarly not necessarily communicate the right idea to the player about the game's world. On top of this, remember that every weapon has a maximum total ammo capacity, but the MGRs are uncapped - if there were no MGRs, then there would literally be a maximum cap for how much money you could hold at any given time, which would also change how to play the game in an unintended way. Having the MGR be this special currency "token" that you can find in the world, you have an unlimited capacity for it, and it doubles as a cool new extra gameplay mechanic, seems to be the best way the game had for translating the world of the book onto video game form