IE, games that don't make up words or names in languages belonging to the game-verse, but borrow as exactly as possible from history, folklore, etymology, and mythology.
(What is this type of game called anyway?)
GURPS has a world called Yrth that has cultures that were sucked in during an event known as the Banestorm (which is the title of the book). The languages and people are essentially "borrowed" from other cultures, and are placed into a fantasy context. It's an interesting concept, and I'd highly recommend it if you're looking for something a little more fantastic, but still grounded in reality. One of the features is that there are faiths from our own world (Christianity, Islam, Judaism), but transposed into another world. Of course there is still some history that's invented for the game world, but there is still the shared culture that preceded it.
Here's a link to the book: http://www.sjgames.com/gurps/books/banestorm/
GURPS 3rd edition also has a lot of alternate history books, as well as several "Who's Who", detailing persons of interest from different cultures. And if you're looking for decently researched historical settings, there's plenty of those to be mined from, such as the Aztecs, Greece, Egypt, Russia, etc..
More info at:
There's also Dark Millennium from BTRC for EABA. It's written from the perspective that Armageddon happened at 1033 AD, and the world continues to exist from that point. It's a pretty fascinating, and I'm looking to really read into this one.
One last one is Deryni Adventures for Fudge, based on the Deryni novels by Katherine Kurtz. The history maps very closely with Middle-ages Europe, and the predominant religion is essentially Catholicism. I'm not sure this is exactly what you're looking for, but it could prove to be an interesting take.
Hope this helps!
An excellent example of a modern day RPG that fits this description is Unknown Armies. Intended as a vehicle for contemporary urban occult games, it supports street level campaigns through to global and even cosmic level campaigns (and the seemless transition between them).
In the past few years, I've played in a couple of long running UA games. One, somewhat like a cross between Naked Lunch, The Matrix and Dark City, is outside the scope of this question, but the other, Election! is right at home here.
In Election! we all played people with an interest in getting the Prime Minister of the UK out of power and in the first session we were told that the PM was an Avatar of the True King, trying to supplant Churchill as the new controller of the archetype (just as Churchill, the British Statesman, supplanted King Arthur, the British King). This would have enormous consequences for the future of democracy in the UK (a new dark age of totalitarian Fortress Britain). We soon discovered that several of the PMs ministers had connections with the Occult underground and were helping him to ascend.
I played a videomancer (one of the standard adepts) and was the bedrock of the media side of our campaign. My favourite Formulae (spell, of sorts) was "Did you threaten to overrule him?" which allowed him to force an interviewee to answer a politically dangerous question truthfully (named from the famous exchange between Jeremy Paxman and Michael Howard on Newsnight). We also had other Avatars and Adepts created especially for the campaign (which is easy to do in UA).
We had lots of fun playing with politics, incorporating current affairs into our game, using our adept powers to alter the course of the election, working out things we could do to gain charges (the magical power Adepts need in order to activate Formulae), our Avatars acting in accordance with their archetypes so that they could gain power and forcing the PM into situations where he had to act against his archetype, so that he lost magical power.
All in all, it was a lot of fun, and it has been surprising in the few years since the game, just how many of the things we did in game have actually happened in real life, UK politics.
As a means to their personal ends, several of us managed to acquire a major charge from the campaign, and my character used his charge to grant himself immortality until all state brutality was ended. *8')
Montsegur 1244. You play Cathars in a city under siege by angry Catholics. When the game ends, everyone has chosen to flee like a coward, convert to Catholicism, or die in a fire as a heretic.
Grey Ranks. You play one of the young Boy Scouts or Girl Guides in 1944 Warsaw. You have to balance the needs of a teenager against the violent environment of war-torn Poland.
AGON. You play Greek heroes, vying for glory as they face the tests of the gods in the ancient world.
Testament (d20 System). You play heroes inspired by Biblical myths in the ancient world (lots of speculative magic).
Zenobia. Historical fantasy inspired by the late Roman Empire (ca. 260 AD). Free!
Ars Magica is the pride of its fans for this very reason—its core and supplements are founded on the principles of science, magic, and mythology as were believed by the people of the middle ages. The basic premise of the game is that all the things those people believed about the world are true: demons, humours, goblins, fairy circles, aether, etc. As a side effect, the "historical authenticity" of the game is very high.
Another is arguably Call of Cthulhu (and the Gumshoe-based Trail of Cthulhu). The stuff in the game and adventures is all drawn from historical fact and from the established Lovecraftian Mythos. The faithfulness of the material to the period is one of the primary indicators of the quality of a supplement to fans.
I don't think this kind of genre has an established, recognised name, but in analogy to book genres I would call it "historical fantasy".
King Arthur Pendragon falls squarely and unashamedly in this area. Its all about replaying the literature around the legends. Everything has its source in the history of Britain and the legends of King Arthur.
I suppose Fiasco could be an example, but it's not so much "borrowing as exactly as possible from..." as "set in this world and time".