I'm not sure what the official title for this type of game but I know it when I see it. Forerunners to this concept include Lords of Creation and maybe even Well of Souls in that they included concepts for characters and environments that were previously considered separate. Starting in 1988 we had moved a step closer with GURPS third edition which got very popular and touted as one of its selling points the ability to mix genres - Magic, Fantasy, Space, Supers, etc. There wasn't any particular book telling you how to do this but it was discussed among players and DM's quite a bit.
Hero System went to a 4th edition in 1989 with the express purpose of making this kind of mixing and matching possible. Pretty soon we had a new Fantasy Hero, Star Hero, and Ninja Hero written as supplements to the main rules instead of as separate games by themselves.
Both games however remained a sort of framework on which you could hang multiple genres if you choose to do so but most actual campaigns were straight-up Fantasy or Sci-Fi or a Supers or a conversion of some other game to these rules. "Universal" is not the same as "Multiversal".
The next logical step was to do this crossover thing as a part of the game setting from the start. This gave us some really cool games that took this idea in wildly different directions.
Shadowrun launched in 1989 and was really the first - "D&D with Guns" was one way to put it. The whole cyberpunk setting with traditional fantasy elements dropped in was a revelation as it did not really reflect any particular novel or movie or comic book that was popular at the time - even though it borrowed from a bunch of existing stuff it truly felt unique among RPG's and pretty much everything else. It had unique mechanics to go with it and they were pretty cool in play too. There was a second edition in 1992 that fixed a lot of small problems with the game, a 3rd edition in 1998 that was still pretty similar, and then a 4th edition in 2004 that went in a somewhat different direction mechanically. I stayed with it from 1st to 3rd, playing and running in multiple campaigns. I know in local circles at least this was maybe the closest to D&D in the ease of getting a game together as everyone could find something to play - magic-user, shaman, ranged combat guy, close-in combat guy, primitive, hi-tech, stealth expert, vehicle pilot- it covered a lot of ground and I never had a problem with a player finding something they wanted to play.
The background (which I assume most people know by now) is a baseline cyberpunk society recovering from a worldwide internet crash when the world is suddenly invaded by the Player's Handbook. Suddenly walking around town with a katana makes sense, especially if you have wired reflexes, body plating, and a friendly mage nearby to heal you. You can get claws like Wolverine, guns like Neo, and throw spells like Dr Strange, although probably not all in one character as magic and cyberware tend to not get along.
It used a dice pool mechanic, the "System of the 90's" much like percentiles were the "System of the 80's" and it worked, although it drove math people crazy trying to calculate probabilities. It was unique and flavorful and really seemed to fit the universe, kind of like the d6 Star wars game seemed to really fit the universe.
In 1990 we got Torg from West End Games which used a system similar to DC Heroes (a good start for this kind of game) which got fairly popular at first due to the spectrum of options and some innovative mechanics but it seemed to fizzle out (locally at least) by the mid-90's
The background was a baseline modern earth that was invaded by other dimensions that would then impose their own laws of reality on part of the earth. North America went Stone Age + Dinosaurs, England went High Fantasy, Japan and Italy went Cyberpunk (2 flavors), and Egypt went 30's Pulp. There were other zones too but you get the idea - if there was a genre that you liked there was probably a place for it in Torg Earth. PC's had a special rule that separated them from normal people in that they could resist the reality-warping effects of the different zones so that the cyberguy's guns would still work in England and magic would still work in the Cyberpapacy.
The mechanics innovation was twofold - something similar to the MEGS system of interchangeable units for time/distance/weight/etc. and the Drama Deck. The deck was pretty cool and dropped random conditions into encounters and could also be used by the players to change things up, sort of like a hero point mechanic.
Now I never played a lot of Torg but I played a few one-offs and liked it enough to pick up some of the game later on. Most of my players never tried it and have no interest in doing so now so I may never run it or play it again but it has a degree of cool that still differentiates it from other games. The flood of supplements turned a lot of people off as they were frequent, sometimes badly edited, and full of changes to the base rules - kind of a triple-whammy when it comes to expansion material.
Also in 1990 we got a pretty heavy hitter in RIFTS. Few can deny the coolness of Rifts' setting and concepts, but even fewer will defend its mechanics as they are ...quirky... to put it kindly. It's the Palladium system which is to say it's a hodge-podge of D&D mechanics (3-18 ability scores, d20 combat resolution), percentile skill system similar to RQ, T2K, and Star Trek, and a point system for magic and psionic powers. Let's just say the mechanics are at least familiar to most players. The background and setting though is all new and incredibly rich.
The baseline is a future-tech earth, similar to early Gamma World, say 100-200 years in the future with the expected technological advances - robots, energy weapons, power armor, cybernetics, biological enhancements, brain implants, etc. Despite these advances someone starts a shooting war and things quickly escalate. As millions die in minutes due to nuclear and other weapon exchanges the psychic energy released rushes around the world reactivating ancient energy pathways (circuits in a way) that circle the earth and where these lines cross they begin to tear open doorways to other dimensions - the RIFTS of the title - and hostile creatures and weird energies come rushing in, killing even more people. Things do eventually stabilize but the population is maybe 10% of what it was and after what is basically a human sacrifice on a planetary scale the earth is supercharged with mystic energy and is now a dimensional nexus unique in the cosmos and things begin to come a callin'. One hundred years later, man's civillization is cast in ruins and a new world rises from the old...
One major city survives in North America and becomes home to a human-supremacist anti-mutant hi-tech power and their skull-themed vehicles and uniforms figured prominently in the artwork and ads for the game and it was pretty cool. The early supplements included Vampire Kingdoms (Mexico is full of vampires) and Atlantis (a fantasy-fest magitech city run by evil alien intelligences who have magical power armor and rune weapons and all kinds of cool stuff. Triax was another early good one featuring Germany as a corporate state using power armor and vehicles to fight off a gargoyle invasion (gargoyles are a type of lesser demon in Rifts).
A potential weakness that only fully came to light later was that every possible thing a character could do is written up as a separate class and there is no multi-classing! However the classes are not a rigid list of powers like D&D but more of a list of bonuses, a few abilities, and a selection of skills followed up by a gear allowance. So they aren't terribly restrictive and the good thing is that every time I have seen it in action every player had a hard time choosing one type of character to run as there are a lot of cool choices. Not many other games cover the range of character types that Rifts does - Dragon Hatchling, Power Armor Pilot, 3 kinds of mage, 3 kinds of psyker, cyborg soldier, crazy (chip-enhanced ninja type), juicer (super-steroid soldier type), Cyber-Knight (Jedi), and Rogue Scientist (Indiana Jones) are just a part of it and each book adds more. The game also allows for the use of their other games to make characters - Ninja Turtles and Superheroes fit in pretty easily as do Palladium Fantasy characters and even Robotech if you have that book and have run out of Zentradi to shoot.
The biggest mechanical problem with the game for some people is the lack of balance between classes, powers, and gear. Compared to something like D&D 3E where balance is a goal and D&D 4E where balance is required, it's jarring - "Who would play a Rogue Scientist or a Wilderness Scout when they can play a Glitter Boy (super power armor with giant gun) as a starting character/" For those of us raised in an earlier era this is not a deal-breaker (most of the time) but it was an issue from Day 1 and it still is as Rifts is using the same system today that it did in 1990 and makes no apologies for it. It adheres to the earlier concept that role-play limitations can balance mechanical advantages that is rejected in most current game systems. If you can live with that, Rifts can be a lot of fun. If not it's an exercise in frustration.
Rifts has rolled on for over 20 years with a steady stream of new material, not all of it good, in the same 2-column black and white softbound format. They occasionally revise an old supplement or put out a compilation of something, but someone who had not looked at the game in 10 years can pick up a new book and understand it just fine. While sometimes derided as a munchkin game, it was very popular in the 90's and has outlasted a lot of other more "mature" or "elegant" games. It is a throwback in some ways to the early days of D&D/AD&D as any ongoing game is guaranteed to have house rules, shared conventions, and possibly even misunderstandings unique to that group - it's the most old-school game still in production today and I admit I have a huge soft spot for it even now.
Since 1990 there really hasn't been another new multiversal game published. I think they are an artifact of the time but I'm not sure why. Underground was a sort of Supers-Cyberpunk combination. Vampire got popular during this time and sucked some of the life out of everything else. I think a lot of people home-brewed it with Hero and GURPS. Maybe the idea was just never that attractive to as many players as it seemed at the time, at least once the initial "Cool!" wore off. Maybe the initial concept that allows the mixing of genres also tends to limit them in some way (it's hard to mix cyber and magic in Shadowrun even if a player really wants to do it) so that it turns off some people who are interested in the idea and they go off and make their own. I'm still not sure it's juts odd to me that while 2 of these 3 games are still going 20 years later the genre of "multigenre" seems almost as dead as the PA and Cyberpunk genres: A few supported games mostly played by the same people that were playing them 10 years ago and not much new in either products or players coming along.