Depending on who you ask, the Tyranids (pictured right) may have inspired the Trope Namers. Their basic infantry, Termagaunts wielding symbiotic guns and bounding, clawed Hormagaunts, are pretty vulnerable to anti-personnel fire, but are dirt-cheap, come in big swarms, and when near a Synapse Creature will continue to surge forward regardless of casualties. With certain rules they can even get the Without Number upgrade, allowing the player to "recycle" dead units, or you can bring along a Tervigon to replenish losses. Either this tide of chitin will overwhelm their opponent, or it will distract them until the Carnifexes, Trygons and other living siege engines smash into the enemy line.
In the background, the Tyranid Hive Mind employs this tactic so often that some Tyranids are born without digestive systems, as they're only intended to fight a single battle and force the enemy to expend ammunition before the real attack. It's also happy to use the "clog the enemy cannons with wreckage" strategy from Futurama, just substituting biological debris for metal.
Interestingly, the other popular Tyranid form is a complete inversion of Zerg Rush called "Nidzilla," where big monsters make up the backbone of an Elite Army. A Nidzilla army typically has very few models on the table, and as many of them as possible are big, bad beasties.
In-Universe, the phenomenon known as the "Shadow in the Warp" that precedes a Tyranid invasion drowns out all psychic communication with the planet, effectively isolating it from anything that requires a Warp Jump to get to it. This "Shadow" is in fact the trillions upon trillions of minds that make up the Hive Mind screaming in unison, essentially making it mental static turned up to 11. For non-blank people, this generally manifests as a mild annoyance. To tuned Psykers, this is enough to drive some of them insane as the thoughts of a galaxy-wide swarm assaults their mind with endless screams of hunger.
This is a perennially-popular Ork tactic as well. Your basic Ork is about as effective in close combat as a Khorne Berserker (sans the Powered Armor), but is only half as expensive, and comes in much bigger mobs - and so long as there's a good number of Orks in that mob, it probably won't break or fall back, especially if led by a Nob. With enough boyz on the table, the enemy won't be able to stop at least some of them from getting into assault range, and it's all downhill from there.
It also helps compensate for other Ork drawbacks. Orks are terrible at ranged fire, to the point where a single Space Marine will be landing exactly twice as many shots as a single Ork. However, an Ork is about half the price of a Space Marine. Mathematically, two-thirds of 20 shots is exactly the same as one-third of 40 shots.
This is shown to be what keeps the Orks going canonically; they lack intelligence, strategy, and suitable technology compared to their foes, but an Ork Waaagh can cover a planet in an ocean of Greenskins. Additionally, any time an Ork dies, which is often, they let off spores which seep into the ground and later grow even more Orks. Sans using fire weapons to burn the spores, their hordes are literally infinite.
The Eldar also can fit under this trope. Their basic unit, the Guardian, is among the weakest of the entire game, and armed with a basic weapon which more-or-less shoots small shurikens at opponents. What it lacks in stopping power and accuracy it makes up for with a ridiculous rate of fire, and a unit of several Eldar Guardians together can completely clog up an enemy target with a single attack, showering them in projectiles.
Indeed, many Imperial commanders have built their careers on throwing Guardsmen at a problem until it went away. Colonel Chenkov once filled gaps in a hastily-constructed wall by executing entire squads of his soldiers, and took a fortress without siege support in a battle that cost him ten million men but won him a medal. He has the "Send In the Next Wave!" special rule that lets him bring in a fresh Conscript Platoon each turn.
Speaking of the Imperial Guard's tanks, the army is unique for being able to take its heavy armor in squadrons, allowing it to bring dozens of tanks and infantry fighting vehicles to the table when other armies might be able to bring three tanks and a half-dozen transports at most.
Though canonically notorious for being a quantity-over-quality army, they do have some of the best defensive capabilities of any faction. An entrenched squad of Imperial Guard on its own can be very hard to take out, due to how much firepower they can exert in such a short amount of time.
More generally, models with the Swarm special rule operate on this principle, and are represented with bases containing multiple small figures. They're particularly vulnerable to flamethrowers, explosives and the like, but each "model" typically has three times as many Wounds as other units and tends to be Fearless, allowing them to at the very least tie up an enemy squad in close combat for a couple of turns. Some of them approach Lethal Joke Character status, like Necron Scarab Swarms that are able to dismantle enemy vehicles, or Nurglings that gain deadly poisoned attacks when special character Epidemus is around.
Warhammer rewrote its rules to accommodate this strategy. Previously only the first rank of a block of infantry could fight at full capacity, with the second or third ranks pitching in if they had spears or pikes, but in its 8th edition the game introduced the Horde rule, allowing units to make ranks of 10 with the first three fighting, and the first two at full capacity - or in other words three times as many models could fight as before.
In terms of particular armies, Orcs and Goblins (and the White DwarfGnoblar and Snotling army lists) tend to be Zerg Rushing factions for the same reasons as in 40k. The Skaven stand out for not only having the cheapest infantry in the game, but for uniquely being able to fire upon their own men while they're locked in close combat thanks to the Life is Cheap rule.
Vampire Counts also abused this mechanic. On top of having the second most powerful combat characters in the game (second only to Chaos Lords and Greater Daemons) they could also raise more undead troops to assist them. Unlike other instances of this ability, the Vampire Counts could raise a unit beyond it's starting limits. With The End Times supplement that brought in Nagash, this took on ridiculous levels as you could create entire regiment's worth of walking corpses on the cheap, while still having enough to buff existing troops or heal them. The only reason they weren't remembered as the horde army was because Skaven brought more from the beginning.
The Soviet Union's primary strategy in Axis And Allies is to build half a dozen infantry or more each turn. However they are only good for defense, Germany has enough industrial capacity to do this with tanks.
This is the last resort of kobolds. As Races of the Dragon and other sourcebooks explain, kobolds know they're individually pretty weak, and are therefore Combat Pragmatists who prefer to wear down attackers through traps and ambushes. If they have no other option, or their traps have failed and their homes are threatened, the kobolds' collectivist mentality leads the men to throw themselves against the enemy by the hundreds, in order to buy time for their women and children to escape with the eggs and ensure the survival of the community.
However, it is the first resort of fiendish armies during Blood War battles, who may have invented. Typically, both demons and devils send hordes of near-mindless grunts against each other, hoping a few might get through to weaken the more capable soldiers. It's not uncommon for both sides to see almost 90% casualties, and it rarely ever accomplishes much, but then, neither has the War itself.
In BattleTech, this was part of the in universe purpose of the Protomech, which was smaller and cheaper than standard Battlemechs while faster and more heavily armed than Battle Armor, allowing them to be built and fielded in large numbers. In gameplay, it's sometimes invoked by flooding the map with dozens of super-cheap infantry or light vehicles, though this is typically considered to be extremely bad sportsmanship due to the Game Breaker nature of the tactic.
An in-universe example of the tactic failing badly are one Mercer Ravannion's attempts to employ "horde tactics" by overrunning enemy forces with swarms of 20-ton 'Mechs like Stingers and Wasps (pretty much the bottom of the weight range) around the late 3010s/early 3020s...which canonically never seem to have actually succeeded at much beyond driving up his own side's casualty count against heavier opposition.
A common beginners' strategy in Risk is to gather up as many soldiers as possible in one country and go on a warpath of Curb Stomp Battles. It is possible to overcome, but it's not easy.
In Strike Legion, this is the Imperium's tried and true tactic. With three million systems under their control, a fleet of fifty million warships, and countless trillions of bodies to throw at the target, they've been wearing down the vastly smaller but vastly more-advanced and better-trained Star Republic through raw numbers. One quote from the Empress herself casually has her order an additional million ships to the front line. At the same time, the Empress has recognized that relying purely on sheer numbers is still inefficient, and has ramped up production of more advanced ships, frames, and Super Soldiers to match the Republic in quality as well as quantity.
Even with all the powers of Caine on their side, the Vampires of World of Darkness know if the masquerade were ever broken and humanity learns of their existence, they would be wiped out in no time.
While almost every Villain in Sentinels of the Multiverse have minions to some degree, a few of them use them and Zerg Rush to their advantage
Omnitron, as he cannot damage the heroes without his toys, relies on getting as many Components and Drones out. His "Self-Aware Robotiscs Factory" side lets him recover a Component or Drone from his trash each turn, while "Rampaging Robot" lets him play a second card each turn. It's pretty hit or miss, as he could just get a Compnent or a One-shot, but if the heroes are lagging behind and going for him instead of the Drones, he can swarm them.
Cosmic Omnitron on the other hand is much better at Zerg Rushing: His "Sentient Dropship" side lets him play the top card of his deck every time he plays a Drone. If he gets lucky, he can swarm them with Drones.
Grand Warlord Voss is a better Zerg Rusher than Omnitron. His entire deck is filled with targets, and his whole strategy relies on overwhelming the heroes with them. Ever better, his Forced Deployment card lets him revive every Minion the heroes kill, so it the heroes get a bunch at ones, they are looking at a very painful round.
The best Zerg Rusher in Sentinels is probably The Matriarch. Every time a foul enters play from her deck, she plays the top card of her deck. Every time. One foul could easily turn into seven.
The Dreamer can also turn into a Zerg Rush once she flipped to "Roused From Slumber" side. On this side, she gets extra card plays based on how many heroes she's fighting.
In X-Wing Miniatures, one fairly common Imperial archetype is to simply buy as many TIE Fighters as you can fit in the points value of the game. TIEs are much less powerful than X-wings and other Rebellion vehicles, but they're also considerably cheaper, to the point where the starter kit can provide two TIEs and one X-Wing and produce a fair fight.